Cyclic error compensation in interferometry systems

ABSTRACT

A first portion of a beam including a first frequency component is directed along a first path. A second portion of the beam is frequency shifted to generate a shifted beam that includes a second frequency component different from the first frequency component and one or more spurious frequency components different from the first frequency component. At least a portion of the shifted beam is directed along a second path different from the first path. An interference signal S(t) from interference between the beam portions directed along the different paths is measured. The signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the paths, where n is an average refractive index along the paths, {tilde over (L)}(t) is a total physical path difference between the paths, and t is time. An error signal is provided to reduce errors in an estimate of {tilde over (L)}(t) that are caused by at least one of the spurious frequency components of the shifted beam, the error signal being derived at least in part based on the signal S(t).

BACKGROUND

This invention relates to interferometers, e.g., displacement measuring and dispersion interferometers that measure displacements of a measurement object such as a mask stage or a wafer stage in a lithography scanner or stepper system, and also interferometers that monitor wavelength and determine intrinsic properties of gases.

Displacement measuring interferometers monitor changes in the position of a measurement object relative to a reference object based on an optical interference signal. The interferometer generates the optical interference signal by overlapping and interfering a measurement beam reflected from the measurement object with a reference beam reflected from the reference object.

In many applications, the measurement and reference beams have orthogonal polarizations and different frequencies. The different frequencies can be produced, for example, by laser Zeeman splitting, by acousto-optical modulation, or internal to the laser using birefringent elements or the like. Some frequency shifting techniques are based on phase modulation. For example, Serrodyne modulation can be applied to a beam using an electro-optic modulator to impose a ramp (or “sawtooth”) phase modulation corresponding to a desired frequency shift. The orthogonal polarizations allow a polarizing beam splitter to direct the measurement and reference beams to the measurement and reference objects, respectively, and combine the reflected measurement and reference beams to form overlapping exit measurement and reference beams. The overlapping exit beams form an output beam that subsequently passes through a polarizer. The polarizer mixes polarizations of the exit measurement and reference beams to form a mixed beam. Components of the exit measurement and reference beams in the mixed beam interfere with one another so that the intensity of the mixed beam varies with the relative phase of the exit measurement and reference beams. A detector measures the time-dependent intensity of the mixed beam and generates an electrical interference signal proportional to that intensity. Because the measurement and reference beams have different frequencies, the electrical interference signal includes a “heterodyne” signal portion having a beat frequency equal to the difference between the frequencies of the exit measurement and reference beams. If the lengths of the measurement and reference paths are changing relative to one another, e.g., by translating a stage that includes the measurement object, the measured beat frequency includes a Doppler shift equal to 2vnp/λ, where v is the relative speed of the measurement and reference objects, λ is the wavelength of the measurement and reference beams, n is the refractive index of the medium through which the light beams travel, e.g., air or vacuum, and p is the number of passes to the reference and measurement objects. Changes in the relative position of the measurement object correspond to changes in the phase of the measured interference signal, with a 2π phase change substantially equal to a distance change L_(RT) of λ/(np), where L_(RT) is a round-trip distance change, e.g., the change in distance to and from a stage that includes the measurement object.

Unfortunately, this equality is not always exact. Many interferometers include nonlinearities such as what are known as “cyclic errors.” The cyclic errors can be expressed as contributions to the phase and/or the intensity of the measured interference signal and have a sinusoidal dependence on the change in optical path length pnL_(RT). For example, a first order harmonic cyclic error in phase has a sinusoidal dependence on (2πpnL_(RT))/λ and a second order harmonic cyclic error in phase has a sinusoidal dependence on 2(2πpnL_(RT))/λ Additional cyclic errors may include higher order harmonic cyclic errors, negative order harmonic cyclic errors, and sub-harmonic cyclic errors.

Cyclic errors can be produced by “beam mixing,” in which a portion of an input beam that nominally forms the reference beam propagates along the measurement path and/or a portion of an input beam that nominally forms the measurement beam propagates along the reference path. Such beam mixing can be caused by ellipticity in the polarizations of the input beams and imperfections in the interferometer components, e.g., imperfections in a polarizing beam splitter used to direct orthogonally polarized input beams along respective reference and measurement paths. Because of beam mixing and the resulting cyclic errors, there is not a strictly linear relation between changes in the phase of the measured interference signal and the relative optical path length pnL between the reference and measurement paths. If not compensated, cyclic errors caused by beam mixing can limit the accuracy of distance changes measured by an interferometer. Cyclic errors can also be produced by imperfections in transmissive surfaces that produce undesired multiple reflections within the interferometer and imperfections in components such as retroreflectors and/or phase retardation plates that produce undesired ellipticities in beams in the interferometer. For a general reference on some theoretical causes of cyclic error, see, for example, C. W. Wu and R. D. Deslattes, “Analytical modelling of the periodic nonlinearity in heterodyne interferometry,” Applied Optics, 37, 6696-6700, 1998.

Another cause of errors, including cyclic errors, is imperfections in the techniques used to impose the frequency shift between the measurement and reference beams. For example, when Serrodyne modulation is used and the sawtooth phase modulation applied to one of the beams has a non-zero fall time and/or an amplitude that is not an exact multiple of 2π, spurious frequency components can result. For examples of some of the effects of imperfect Serrodyne modulation, see Laskoskie et al., “Ti—LiNbO3 Waveguide Serrodyne Modulator with Ultrahigh Sideband Suppression for Fiber Optic Gyroscopes.,” Journal of Lightwave Technology, Vol. 7, No. 4, 600-606, April 1989.

In dispersion measuring applications, optical path length measurements are made at multiple wavelengths, e.g., 532 nm and 1064 nm, and are used to measure dispersion of a gas in the measurement path of the distance measuring interferometer. The dispersion measurement can be used to convert the optical path length measured by a distance measuring interferometer into a physical length. Such a conversion can be important since changes in the measured optical path length can be caused by gas turbulence and/or by a change in the average density of the gas in the measurement arm even though the physical distance to the measurement object is unchanged. In addition to the extrinsic dispersion measurement, the conversion of the optical path length to a physical length requires knowledge of an intrinsic value of the gas. The factor Γ is a suitable intrinsic value and is the reciprocal dispersive power of the gas for the wavelengths used in the dispersion interferometry. The factor Γ can be measured separately or based on literature values. Cyclic errors in the interferometer also contribute to dispersion measurements and measurements of the factor Γ. In addition, cyclic errors can degrade interferometric measurements used to measure and/or monitor the wavelength of a beam.

The interferometers described above are often crucial components of scanner systems and stepper systems used in lithography to produce integrated circuits on semiconductor wafers. Such lithography systems typically include a translatable stage to support and fix the wafer, focusing optics used to direct a radiation beam onto the wafer, a scanner or stepper system for translating the stage relative to the exposure beam, and one or more interferometers. Each interferometer directs a measurement beam to, and receives a reflected measurement beam from, a plane mirror attached to the stage. Each interferometer interferes its reflected measurement beams with a corresponding reference beam, and collectively the interferometers accurately measure changes in the position of the stage relative to the radiation beam. The interferometers enable the lithography system to precisely control which regions of the wafer are exposed to the radiation beam.

In practice, the interferometry systems are used to measure the position of the wafer stage along multiple measurement axes. For example, defining a Cartesian coordinate system in which the wafer stage lies in the x-y plane, measurements are typically made of the x and y positions of the stage as well as the angular orientation of the stage with respect to the z axis, as the wafer stage is translated along the x-y plane. Furthermore, it may be desirable to also monitor tilts of the wafer stage out of the x-y plane. For example, accurate characterization of such tilts may be necessary to calculate Abbe offset errors in the x and y positions. Thus, depending on the desired application, there may be up to five degrees of freedom to be measured. Moreover, in some applications, it is desirable to also monitor the position of the stage with respect to the z-axis, resulting in a sixth degree of freedom.

SUMMARY

Among other aspects, the invention features electronic processing methods that characterize and compensate cyclic errors in interferometric data. Because cyclic errors are compensated electronically, the interferometry system that produces the data has greater tolerance to optical, mechanical, and electronic imperfections that can cause cyclic errors, without sacrificing accuracy. The compensation techniques are especially useful for interferometric data used to position microlithographic stage systems.

In part, the invention is based on the realization that prior values of a main interferometric signal can be used to reduce the effect of cyclic errors on an estimate of a length being measured by the interferometry system (e.g., a length indicating the position of a stage). Any of a number of signal transformations such as a quadrature signal or a Fourier transform can be derived from the prior values of the main interferometric signal. These derived signal transformations can then be used to generate one or more error basis functions that represent one or more cyclic error terms in the main interferometric signal. Appropriate amounts of each error basis function (e.g., as determined by a coefficient for each error basis function) form an error signal that is subtracted from a signal from which the measured length is obtained. The corresponding reduction in cyclic errors increases the accuracy of the measured length. Using the error signal and the resulting compensated length estimate, the position of the measurement object can be monitored and/or controlled in a variety of applications.

For example, the prior values of the main interferometric signal can be used to calculate an estimate for a quadrature signal for the main interferometric signal. Algebraic combinations of such signals can yield error basis functions in the form of sinusoidal functions whose time-varying arguments correspond to particular cyclic error terms. In embodiments in which the interferometer beams have a heterodyne frequency splitting, one may also calculate the quadrature signal of the heterodyne reference signal, and the error basis functions may be derived from algebraic combinations the main signal, the reference signal, and the quadrature signals of the main and reference signals.

The error basis functions are used to isolate particular cyclic error terms in the main signal and characterize coefficients representative of each cyclic error term (e.g., its amplitude and phase). For example, algebraic combinations of the error basis functions and the main signal and its quadrature signal can move a selected cyclic error term to zero-frequency, where low-pass filtering techniques (e.g., averaging) can be used to determine its amplitude and phase. Such coefficients are stored. Thereafter, a superposition of the error basis functions weighted by the stored coefficients can be used to generate an error signal that can be subtracted from the main signal to reduce the cyclic errors therein and improve its accuracy.

The technique is particularly useful when the Doppler shift is small relative to the heterodyne frequency because the frequency of each cyclic error term is nearly equal to that of primary component of the main signal, in which case the estimate for the quadrature signal of the main signal is more accurate. This is an especially important property because it is precisely when the frequencies of the cyclic error terms are near that of the primary component of the main signal that the cyclic error terms are most problematic because they cannot be removed by frequency filtering techniques. Furthermore, at small Doppler shifts, one or more of the cyclic error frequencies may be within the bandwidth of a servo system used to position a stage based on the interferometric signal, in which the case the servo loop may actually amplify the cyclic error term when positioning the stage. Small Doppler shifts are actually quite common in microlithographic stage systems, such as when searching for an alignment mark, scanning in an orthogonal dimension to the one monitored by the interferometric signal, and changing stage direction. Moreover, at small Doppler shifts, selecting an integral relationship between the sampling rate of the detector and the heterodyne frequency (e.g., 6:1) yields an especially simple formula for the quadrature signal.

In addition, at small Doppler shifts, the main signal is nearly periodic with the heterodyne frequency, in which case prior data can be used to generate the error signal. As a result, correction of the main signal can be accomplished with only a single real-time subtraction of the error signal from the main signal, significantly reducing the computation time associated with the correction and thereby reducing data age errors in any servo system for position a microlithography stage.

Error basis functions can be derived from other combinations of prior values of the main interferometric signal besides a quadrature signal including a Fourier transform of the main interferometric signal. In the case of a Fourier transform, the resulting error signal is subtracted from a complex signal from which the measured length is obtained, as described in more detail below. In some cases (e.g., when the stage is moving slowly or is nearly stationary), the error basis functions can be derived from a distribution of values, where each of the values is generated from multiple values of the main interferometric signal, and the values in the distribution (e.g., multi-dimensional values distributed over a multi-dimensional space such as a complex space with real and imaginary dimensions) are not necessarily sequential in time.

We now summarize various aspects and features of the invention.

In one aspect, in general, the invention features a method that includes directing a first portion of a beam including a first frequency component along a first path. The method includes frequency shifting a second portion of the beam to generate a shifted beam that includes a second frequency component different from the first frequency component and one or more spurious frequency components different from the first frequency component. The method includes directing at least a portion of the shifted beam along a second path different from the first path. The method includes measuring an interference signal S(t) from interference between the beam portions directed along the different paths. The signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the paths, where n is an average refractive index along the paths, {tilde over (L)}(t) is a total physical path difference between the paths, and t is time. The method includes providing an error signal to reduce errors in an estimate of {tilde over (L)}(t) that are caused by at least one of the spurious frequency components of the shifted beam, the error signal being derived at least in part based on the signal S(t).

Aspects of the invention can include one or more of the following features.

The method further comprises providing the estimate of {tilde over (L)}(t) based at least in part on subtracting the error signal from the signal S(t).

Frequency shifting the second beam portion comprises modulating the phase of the second beam portion to generate the shifted beam.

The phase modulation comprises electro-optic phase modulation.

The phase modulation comprises Serrodyne modulation.

The errors are dependent on a non-zero fall time of a ramp associated with the Serrodyne modulation.

The errors are further dependent on an amplitude of the ramp.

The method further comprises providing the estimate of {tilde over (L)}(t) based at least in part on adjusting the phase modulation of the second beam portion using the error signal.

The phase modulation is adjusted using the phase of the error signal.

An amplitude of the phase modulation is adjusted using the phase of the error signal.

At least one of the beam portions is directed to reflect from a movable measurement object before producing the interference signal S(t) to generate a Doppler shift indicative of movement of the measurement object.

The method further comprises providing the estimate of {tilde over (L)}(t) based at least in part on a sinusoidal component of the signal S(t) that has a time-varying argument corresponding to a sum of a term proportional to a difference between peak frequencies of the first frequency component and the second frequency component and a term proportional to a Doppler shift frequency.

The error signal includes a sinusoidal component that has a time-varying argument corresponding to a difference between the term proportional to the difference between the peak frequencies of the first frequency component and the second frequency component and the term proportional to the Doppler shift frequency.

The difference between the peak frequencies of the first frequency component and the second frequency component is less than about 500 kHz.

The Doppler shift frequency is less than about 50 kHz.

One of the paths is associated with a position of a reference object and the other path is associated with a position of a moveable measurement object.

The position of the moveable object is controlled by a servo system.

The servo system controls the position of the measurement object based on the signal S(t) and the error signal.

Providing the error signal to reduce the errors caused by at least one spurious frequency component in the shifted beam comprises providing one or more coefficients representative of one or more errors that cause the signal S(t) to deviate from an ideal expression of the form A₁ cos(ω_(R)t+φ(t)+ζ₁), where A₁ and ζ₁ are constants, ω_(R) is an angular frequency difference between the first frequency component and the second frequency component, and φ(t)=nk{tilde over (L)}(t), with k=2π/λ and λ equal to a wavelength for the beam.

The deviation can be expressed as

${\sum\limits_{m,p}{A_{m,p}{\cos\left( {{\omega_{R}t} + {\frac{m}{p}{\varphi(t)}} + \zeta_{m,p}} \right)}}},$ where p=1, 2, 3 . . . , and m is any integer not equal to p, and where the provided coefficients comprise information corresponding to at least some of A_(m,p) and ζ_(m,p).

The coefficients are derived at least in part based on a plurality of samples of the signal S(t).

The error signal is generated from the coefficients and one or more error basis functions derived at least in part from a plurality of samples of the signal S(t).

Each of the error basis functions is derived at least in part from a linear combination of samples of the signal S(t).

Each of the error basis functions corresponds to a function that includes one or more leading sinusoidal terms having a time-varying argument that corresponds to a time-varying argument of an error term that represents a portion of the deviation of S(t) from the ideal expression.

Reducing errors in the estimate of {tilde over (L)}(t) comprises deriving the estimate of {tilde over (L)}(t) from a difference between the error signal and a discrete Fourier transform of samples of S(t).

A lithography method for use in fabricating integrated circuits on a wafer includes supporting the wafer on a moveable stage; imaging spatially patterned radiation onto the wafer; adjusting the position of the stage; and monitoring the position of the stage using an interferometry system, wherein monitoring the position of the stage comprises reducing errors in an estimate of a physical path difference associated with a position of a measurement object associated with the stage using the method described above.

A method for fabricating integrated circuits includes applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography method described above; and producing an integrated circuit from the wafer.

A lithography method for use in the fabrication of integrated circuits includes directing input radiation through a mask to produce spatially patterned radiation; positioning the mask relative to the input radiation; monitoring the position of the mask relative to the input radiation using an interferometry system, wherein monitoring the position of the mask comprises reducing errors in an estimate of a physical path difference associated with the position of the mask using the method described above; and imaging the spatially patterned radiation onto a wafer.

A method for fabricating integrated circuits includes applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography method described above; and producing an integrated circuit from the wafer.

A lithography method for fabricating integrated circuits on a wafer includes positioning a first component of a lithography system relative to a second component of a lithography system to expose the wafer to spatially patterned radiation; and monitoring the position of the first component relative to the second component using an interferometry system. Monitoring the position of the first component comprises reducing errors in an estimate of a physical path difference associated with a position of a measurement object associated with the first component using the method described above.

The first component comprises projection optics, and the second component comprises a stage for supporting the wafer.

A method for fabricating integrated circuits includes applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography method described above; and producing an integrated circuit from the wafer.

A method for fabricating a lithography mask includes directing a write beam to a substrate to pattern the substrate; positioning the substrate relative to the write beam; and monitoring the position of the substrate relative to the write beam using an interferometry system. Monitoring the position of the substrate comprises reducing errors in an estimate of a physical path difference associated with a position of a measurement object associated with the substrate using the method described above.

An apparatus comprising a computer readable medium which during operation causes a processor to perform the method described above.

In another aspect, in general, the invention features an apparatus that includes an interometry system. During operation, the interometry system directs a first portion of a beam including a first frequency component along a first path; frequency shifts a second portion of the beam to generate a shifted beam that includes a second frequency component different from the first frequency component and one or more spurious frequency components different from the first frequency component; and directs at least a portion of the shifted beam along a second path different from the first path. The apparatus includes a detector that measures an interference signal S(t) from interference between the beam portions directed along the different paths, wherein the signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the paths, where n is an average refractive index along the paths, {tilde over (L)}(t) is a total physical path difference between the paths, and t is time. The apparatus includes an electronic processor, which during operation receives the interference signal S(t) from the detector and provides an error signal to reduce errors in an estimate of {tilde over (L)}(t) that are caused by at least one of the spurious frequency components of the shifted beam, the error signal being derived at least in part based on the signal S(t).

Aspects of the invention can include one or more of the following features.

The second beam portion is frequency shifted in a phase modulator that modulates the phase of the second beam portion to generate the shifted beam.

The phase modulator comprises an electro-optic phase modulator.

A lithography system for use in fabricating integrated circuits on a wafer includes a stage for supporting the wafer; an illumination system for imaging spatially patterned radiation onto the wafer; a positioning system for adjusting the position of the stage relative to the imaged radiation; and the apparatus described above for monitoring the position of the wafer relative to the imaged radiation.

A method for fabricating integrated circuits includes applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography system described above; and producing an integrated circuit from the wafer.

A lithography system for use in fabricating integrated circuits on a wafer includes a stage for supporting the wafer; and an illumination system including a radiation source, a mask, a positioning system, a lens assembly, and the apparatus described above. During operation the source directs radiation through the mask to produce spatially patterned radiation, the positioning system adjusts the position of the mask relative to the radiation from the source, the lens assembly images the spatially patterned radiation onto the wafer, and the apparatus monitors the position of the mask relative to the radiation from the source.

A method for fabricating integrated circuits includes applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography system described above; and producing an integrated circuit from the wafer.

A beam writing system for use in fabricating a lithography mask includes a source providing a write beam to pattern a substrate; a stage supporting the substrate; a beam directing assembly for delivering the write beam to the substrate; a positioning system for positioning the stage and beam directing assembly relative one another; and the apparatus described above for monitoring the position of the stage relative to the beam directing assembly.

A method for fabricating a lithography mask includes directing a beam to a substrate using the beam writing system described above; varying the intensity or the position of the beam at the substrate to form a pattern in the substrate; and forming the lithography mask from the patterned substrate.

As used herein “algebraic combinations” means combinations of operands (e.g., real or complex numbers including values of signals) according to one or more algebraic operations (e.g., addition, subtraction, multiplication, and division).

Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. In case of conflict with publications, patent applications, patents, and other references mentioned incorporated herein by reference, the present specification, including definitions, will control.

Other features and advantages of the invention will be apparent from the following detailed description.

DESCRIPTION OF DRAWINGS

FIG. 1 is a block diagram of an interferometric position measuring system.

FIGS. 2 a and 2 b are plots of a phase modulation and a resulting interference signal, respectively.

FIGS. 3 a-3 d are plots of a spectrum of an interference signal.

FIG. 4 a is a schematic diagram of a processing unit for generating cyclic error basis functions and characterizing cyclic error coefficients based on a main interference signal S(t) and a reference signal S_(R)(t).

FIG. 4 b is a schematic diagram of a processing unit for generating an error signal S_(ψ)(t) from the cyclic error basis functions and characterized coefficients and using the error signal to reduce cyclic errors in the main interference signal S(t).

FIG. 4 c is a schematic diagram of an exemplary measurement system for generating cyclic error basis functions and characterizing cyclic error coefficients based on a complex measurement signal.

FIG. 4 d is a schematic diagram of an error estimator for the measurement system of FIG. 4 c.

FIG. 4 e is a schematic diagram of a processing unit for the error estimator of FIG. 4 d.

FIG. 5 is a schematic diagram of an M^(th) order digital filter for use in low-pass filtering algebraic combinations of the main signal, the reference signal, their quadrature signals, and the error basis functions to yield the cyclic error coefficients.

FIG. 6 is a schematic diagram of an interferometry system including a high-stability plane mirror interferometer (HSPMI).

FIG. 7 is a schematic diagram of an embodiment of a lithography tool that includes an interferometer.

FIG. 8 a and FIG. 8 b are flow charts that describe steps for making integrated circuits.

FIG. 9 is a schematic of a beam writing system that includes an interferometry system.

DETAILED DESCRIPTION

Referring to FIG. 1, an exemplary interferometric position measurement system 1 includes a source 2 (e.g., a laser) that provides a first input beam whose spectrum has a frequency component with a narrow linewidth and a peak frequency corresponding to the optical wavelength of the beam (e.g., 1550 nm). A frequency shifter 4 frequency shifts a portion of the first input beam to generate a second frequency shifted input beam that includes a frequency component whose peak frequency is shifted relative to the peak frequency of the first input beam. The system 1 also includes an interferometer 6 to provide an interference signal that represents an interferometric measurement of the position of a measurement object, and a detection system 8 to detect and analyze the interference signal to obtain an estimate of the position of the measurement object.

The frequency shifter 4 can use any of a variety of techniques to generate the measurement and reference beams depending, for example, on the magnitude of the desired frequency difference between the beams. In some systems, the beat frequency is on the order of a few MHz or tens of MHz. In some systems, e.g. when the maximum Doppler frequency is low, a lower beat frequency may be desirable. Circuitry operating at lower frequencies may have advantages of cost, noise, or dynamic range. For example, in a system with a maximum Doppler frequency of 20 kHz, a beat frequency on the order of 100 kHz might be preferred. One approach for generating measurement and reference beams that differ by a low beat frequency uses an electro-optic modulator (EOM) for Serrodyne modulation, which applies a sawtooth phase modulation to an input beam to generate the modulated beam. One advantage of using an EOM in such systems is that there are readily available low cost and reliable devices (e.g., optical telecommunications devices) that include useful features such as low-insertion loss optical fiber couplers. In addition to cases in which a low beat and/or Doppler frequency is being used, Serrodyne modulation may also be desirable for other reasons for both low and high frequencies (e.g., beat frequency of a few or tens of MHz or more), for example, ruggedness, reliability, wavelength, solid state laser, low cost of EOMs, etc.

The interferometer 6 accepts the first and second input beams and directs them as measurement and reference beams along measurement and reference paths of the interferometer. Either of the input beams can serve as the measurement beam or the reference beam. The interferometer 6 includes the measurement object whose position is being measured relative to a reference location defined by the interferometer 6. For example, the measurement object can include a mirror from which the measurement beam is reflected at some point along the measurement path.

The detection system 8 detects optical interference between the two beams in the interferometer 6 to generate an electronic interference signal S(t) that can be processed electronically, for example, after digital sampling of the signal. The signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the paths, where n is an average refractive index along the paths, {tilde over (L)}(t) is a total physical path difference between the paths, and t is time. The detection system 8 is able to reduce errors in an estimate of {tilde over (L)}(t) that result from various causes using appropriate techniques. As an example, we will consider errors due to imperfect Serrodyne modulation in the frequency shifter 4 as a cause of errors.

The interference signal S(t) can be expressed as a main sinusoidal term representing a Doppler shifted measurement component plus an additional error term representing one or more error components as: S(t)=A ₁ cos(φ_(R)+φ)+S _(ψ)(t).

The time dependent phase φ_(R) is a reference phase resulting from the frequency difference (or “beat frequency”) between the measurement beam and the reference beam, with the beat frequency f_(R) given by |dφ_(R)/dt|=ω_(R)=2πf_(R). The time dependent phase φ represents the Doppler shift due to movement of a measurement object that is part of the interferometer 6, with the Doppler frequency f_(D) given by |dφ/dt|=ω_(D)=2πf_(D). S_(ψ)(t) is an error term that represents error components including cyclic error components and other error components such as components at or near the harmonic frequencies of the beat frequency.

Referring to FIG. 2 a, the frequency shifter 4 modulates the optical path length (e.g., by electro-optically modulating the index of refraction) to generate a periodic sawtooth phase pattern 100 that has a substantially linear rise and a fast fall over a period T. The amplitude A of the imposed phase shift (in units of radians) is selected to be substantially equal to a multiple of 2π (e.g., within a few percent) A≈2πm. In some cases, a small multiple is selected (e.g., m=1) because the frequency shifter 4 may have a maximum voltage or electrical power rating, or the frequency shifter 4 may have increased nonlinearities at the larger voltage range needed for a larger phase shift, for example. The imposed sawtooth phase modulation results in beat frequency that is then given by m/T. Thus, the beat frequency can be tuned by changing the integer m and/or the period T.

FIG. 2B shows an interference pattern 101 over a period T of an exemplary interference signal S(t) detected from the interferometer 6 driven by an input beam that has been modulated with the phase pattern 100. The slower sinusoidal shape 102 during the rise of the phase pattern 100 corresponds to the desired beat frequency plus a Doppler frequency from movement of the measurement object. The faster sinusoidal shape 103 during the fall of the phase pattern 100 corresponds to “unwinding” over the short fall time of the phase accumulated during the rise, plus the same Doppler shift. This high frequency “flyback” generates spurious frequency components in the spectrum of the interference signal S(t) that appear at multiple frequencies.

FIG. 3A shows a spectrum 80 that corresponds to a sawtooth phase modulation with a zero fall time and an amplitude of A=2π. In this example, the beat frequency is

$f_{R} = {\frac{1}{T} = {100\mspace{14mu}{kHz}}}$ and the Doppler frequency is f_(D)=10 kHz. There is a measurement frequency component 82 at the Doppler shifted measurement frequency of f_(R)+f_(D)=110 kHz.

FIG. 3B shows a spectrum 84 that corresponds to a sawtooth phase modulation with a non-zero fall time of 200 ns and an amplitude of A=2π. The beat frequency and Doppler frequency are the same as in the previous example of FIG. 3A (and in the examples of FIGS. 3C and 3D). In this example, in addition to a Doppler shifted measurement frequency component 86, there are additional “spurious” frequency components due mainly to the flyback portions of the phase modulation. Other imperfections that can affect the spurious frequency components include, for example, EOM drive amplitude value, EOM drive amplifier nonlinearity and EOM nonlinearity. Since the flybacks occur periodically, repeating at time intervals of T, the peak frequencies of these spurious components appear near the beat frequency

$f_{R} = \frac{1}{T}$ and near harmonics of this frequency 2/T, 3/T, 4/T, etc. The peak frequencies also depend on the Doppler shift which determines the actual phase accumulated in the sawtooth period. Thus, a “negative Doppler” frequency component 88 occurs at a frequency of f_(R)−f_(D)=90 kHz, and pairs of frequency components also occur at frequencies

${\frac{n}{T} \pm f_{D}},$ where n=2, 3, 4, etc. A more detailed derivation of similar frequency characteristics is available, for example, in Voges et al., “Optical phase and amplitude measurement by single sideband homodyne detection” IEEE Journal of Quantum Electronics, Vol. 18, pages 124-129, 1982, incorporated herein by reference.

Most of these spurious frequency components can be compensated for using bandpass filtering techniques. For example, the closest of these other frequency components appears at a frequency of

${\frac{2}{T} - f_{D}} = {190\mspace{14mu}{{kHz}.}}$ Even at small Doppler shifts (relative to the beat frequency) due to low speed measurement object movement, there remains a distance of almost f_(R)=100 kHz between the measurement frequency component 86 and this spurious component. However, at small Doppler shifts, the negative Doppler frequency component 88 could be so close to the measurement frequency component 86 that filtering techniques could be ineffective or impractical.

In such cases, the negative Doppler frequency component 88 can be more easily compensated for using the cyclic error compensation techniques described herein. Cyclic errors appear at frequencies that are shifted from the beat frequency by multiples of the Doppler frequency, and can be classified according to the frequency offset from the reference phase φ_(R) as modeled in Equations (3)-(6) below. The negative Doppler frequency component 88 corresponds to a cyclic error of order “−1” with time dependent phase φ_(R)−φ (modeled in Equation (3)). Other cyclic error orders, which are not significant in this example, include order “0” with time dependent phase φ_(R) (modeled in Equation (4)), order “2” with time dependent phase φ_(R)+2φ (modeled in Equation (5)), and order “3” with time dependent phase φ_(R)+3φ (modeled in Equation (6)). Thus, while the techniques described herein for compensating for cyclic errors are capable of compensating for multiple orders of cyclic errors, in this example the single cyclic error of order “−1” is corrected using these techniques. In some systems, there may also be cyclic errors of various orders due to other causes (e.g., optical) that may simultaneously be compensated for by these techniques. By forming an error signal that can be subtracted from the measurement signal, more accurate compensation of the errors affecting the estimate of {tilde over (L)}(t) can be achieved in some cases than can be achieved through techniques like bandpass filtering.

The magnitude of the spurious frequency components can also be reduced by adjusting characteristics of the phase modulation. In addition to reducing the fall time of the sawtooth, the magnitude of the spurious frequency components can be reduced by adjusting the amplitude of the sawtooth A away from an exact 2π phase shift by small amounts (e.g., by a few percent). FIG. 3C shows a spectrum 90 that corresponds to a sawtooth phase modulation with a non-zero fall time of 200 ns and an amplitude of A=2π(1.01). With this increase of 1% in the amplitude A, the magnitude of the largest spurious component 92 is reduced relative to the magnitude of the measurement component 91. FIG. 3D shows a spectrum 94 that corresponds to a sawtooth phase modulation with a non-zero fall time of 200 ns and an amplitude of A=2π(1.04). With this increase of 4% in the amplitude A, the magnitude of the negative Doppler frequency component 96 is reduced relative to the magnitude of the measurement component 95. In this case, it is evident that adjusting the amplitude A to minimize the negative Doppler frequency component 96 comes at the expense of increasing the amplitudes of some of the components at the higher harmonics. However, since the effects of the components at the higher harmonics can be reduced using filtering techniques, it can be useful to minimize the negative Doppler component. Other adjustments to the Serrodyne phase modulation can be made to attempt to further reduce the negative Doppler component perhaps at the expense of other error components, such as adjusting the shape of the ramp of the sawtooth away from perfect linearity, for example.

The use of CEC enables measuring both the magnitude and phase of the error components. The phase, which is related to amplitude A, facilitates direct adjustment of the modulation magnitude to minimize the error magnitudes. The phase of the negative Doppler component may be determined by ζ⁻¹=arg(A⁻¹,B⁻¹) using the values in equations 37 and 38, or equivalently by ζ⁻¹=arg(C_(4R)) using the value in equation 102. This phase will exhibit a 2π phase change as the ramp gain is adjusted from below the optimum value to above the optimum value. Thus, the phase may be used as an indicator of whether to increase or decrease the ramp gain.

Embodiments include an electronic cyclic error compensation (CEC) procedure for compensation of cyclic error effects in interferometry applications, such as heterodyne interferometry. In preferred embodiments, the compensation is achieved for low slew rates of a plane mirror measurement object attached to a stage or attached to a reference system with associated interferometers attached to a stage. When optical techniques are used to eliminate and/or reduce the amplitudes of certain cyclic errors such as sub-harmonic cyclic errors to ≲0.05 nm (3σ), the remaining cyclic errors of the harmonic type with amplitudes of 0.5 nm or less can be treated as having constant amplitudes with fixed offset phases and the required accuracy of the cyclic error compensation for a remaining cyclic error term is approximately 10% to meet a compensated cyclic error budget of 0.05 nm (3σ) or less. Further, the number of cyclic error terms that need to be compensated electronically are typically a small number, e.g., of the order of 3. In preferred embodiments, the processing operations of CEC at high digital processing rates can be limited to a single add operation, whereas the remaining processing operations, which require additions, subtractions, multiplications, and divisions, can be performed at lower rates using prior values of the interference signal.

Typically, cyclic error effects in heterodyne interferometry can be eliminated by filtering the heterodyne signal in frequency space, e.g., using Fourier spectral analysis, when the Doppler shift frequencies can be resolved by the frequency resolution of a phase meter used to determine the heterodyne phase. Unfortunately, such filtering techniques cannot be used to eliminate cyclic error effects at low slew rates of the stage (including, e.g., zero speed of the stage) when the corresponding Doppler shift frequencies cannot be distinguished from the frequency of the primary signal. Further complications arise with cyclic error frequencies are within the bandwidth of the servo system, in which case the cyclic errors can be coupled directly into the stage position through the servo control system, and even amplify the error in the stage position from a desired position.

Specific details of embodiments of the CEC are described further below. In one approach, the CEC procedure processes real time-sampled values of a digitized measurement signal (DMS) generated by an analog-to-digital-converter (ADC). Advantages of this “DMS approach” include a cyclic error correction signal that may be generated in a “feed forward mode,” where the feed forward mode can involve a simple discrete transform based on a translation in time and need not require a spectral analysis or the use of a discrete transform such as a discrete Fourier transform, such as a fast Fourier transform (FFT). Likewise, conjugated quadratures of the main interference signal and the reference signal can be generated by simple discrete transforms and need not require the use of a discrete transform such as a discrete Hilbert transform. Moreover, the feed forward mode can reduce the number of computer logic operations that are required at the very highest compute rates and can thereby reduce errors in data age that are introduced by incorporation of CEC.

In another approach, the CEC procedure processes complex values of a complex measurement signal (CMS) generated by a discrete Fourier transform (DFT) module after the ADC module. Advantages of this “CMS approach” include the ability to update the DFT (and the CEC computations) at a lower rate (e.g., 10 MHz) than the ADC sampling rate (e.g., 120 MHz). A reduction in the CEC update rate enables a simplified hardware architecture. For example, a reduction in CEC update rate by a factor of 12 can result in a hardware savings of greater than a factor of 12. The CMS approach also eliminates cyclic errors that are due to finite arithmetic precision of the samples generated by the ADC and of the DFT coefficients and calculations in the DFT module. The CMS approach is also less subject to noise than the DMS approach due to the number of samples and the window function used by the DFT module.

Another advantage of both the DMS approach and the CMS approach is that the cyclic error coefficients can be characterized at Doppler shift frequencies for which the phase meter cannot distinguish between the cyclic error frequencies from the frequency of the primary component of the interference signal. Furthermore, the cyclic error coefficients can be characterized and used for compensation over a range of Doppler shift frequencies that is small relative to heterodyne frequency, which is a range over which the cyclic error coefficients are typically frequency independent, thereby simplifying the cyclic error correction.

We now describe the DMS approach for the CEC that operates in a feed forward mode of operation, in which a cyclic error correction signal S_(ψ)(t) is subtracted from a corresponding electrical interference signal S(t) of an interferometer to produce a compensated electrical interference signal. The phase of the compensated electrical interference signal is then measured by a phase meter to extract relative path length information associated with the particular interferometer arrangement. Because cyclic error effects have been reduced, the relative path length information is more accurate. As a result, the compensated electrical interference phase can be used to measure and control through a servo control system the position of a stage, even at low slew rates including a zero slew rate where cyclic error effects can otherwise be especially problematic. The DMS approach is also described in published U.S. application Ser. No. 10/616,504 (publication number US 2004/0085545 A1).

Referring to FIGS. 4 a and 4 b, in a preferred embodiment, the CEC comprises two processing units. One processing unit 10 determines cyclic error basis functions and factors relating to the amplitudes and offset phases of cyclic errors that need be compensated. A second processing unit 60 of CEC generates cyclic error correction signal S_(ψ)(t) using the cyclic error basis functions and the factors relating to the amplitudes and offset phases determined by first processing unit 10. The first processing unit 10 of CEC for the first embodiment is shown schematically in FIG. 4 a and the second processing unit 60 of CEC of the first embodiment is shown schematically in FIG. 4 b.

Referring now to FIG. 4 a, an optical signal 11 from an interferometer is detected by detector 12 to generate an electrical interference signal. The electrical interference signal is converted to digital format by an analog to digital converter (ADC) in converter/filter 52 as electrical interference signal S(t) and sent to the CEC processor. For example, the ADC conversion rate is a high rate, e.g., 120 MHz.

In the present embodiment, we focus on a particular set of four cyclic error terms that are compensated at low slew rates. Adaptation to compensate for a different set of cyclic errors will be evident to one skilled in the art based on the subsequent description. The electrical interference signal S(t) comprising the four cyclic error terms can be expressed in the form S(t)=A ₁ cos(φ_(R)+φ+ζ₁)+S _(ψ)(t)  (1) where S _(ψ)(t)=S _(ψ−1)(t)+S _(ψ0) +S _(ψ2)(t)+S _(ψ3)(t);  (2) S _(ψ−1)(t)=ε⁻¹ cos(φ_(R)−φ+Λ⁻¹),  (3) S _(ψ0)=ε₀ cos(φ_(R)+ζ₀),  (4) S _(ψ2)(t)=ε₂ cos(φ_(R)+2φ+ζ₂),  (5) S _(ψ3)(t)=ε₃ cos(φ_(R)+3φ+ζ₃);  (6) φ_(R) is the phase of a reference signal S_(R)(t) with dφ_(R)/dt=ω_(R) corresponding to 2π times the frequency difference of the measurement beam and reference beam components of the input beam to the interferometer; A₁ and ζ₁ are the amplitude and offset phase, respectively, of the primary component of the electrical interference signal; ε⁻¹, ε₀, ε₂, and ε₃ are the amplitudes for the cyclic error terms; ζ⁻¹, ζ₀, ζ₂, and ζ₃ are the offset phases of the cyclic error terms; φ=4kL  (7) for a plane mirror interferometer such as a HSPMI (which involves two passes of the measurement beam to the measurement object); k is a wavenumber corresponding to wavelength λ of beam 11; and L is the difference between the one way physical length of the measurement path and the one way physical length of the reference path of the interferometer. Cyclic error amplitudes ε⁻¹, ε₀, ε₂, and ε₃ are much less than the A₁, i.e. ≲( 1/50)A₁. An example of the frequency difference ω_(R)/2π is 20 MHz.

Note that there is generally a set of cyclic error terms whose phases are independent of φ_(R). This set of cyclic error terms has been omitted from Equation (2) because they are eliminated by a high pass filter in converter/filter 52.

The factors relating to amplitudes ε_(p) and offset phases ζ_(p) of the cyclic error terms and the time dependent factors of the cyclic error terms are generated using measured values of both S(t) and reference signal S_(R)(t). The factors relating to amplitudes ε_(p) and offset phases ζ_(p) are determined and the results transmitted to a table 40 for subsequent use in generation of the cyclic error correction signal S_(ψ)(t). The time dependent factors of the cyclic error terms are obtained by application of simple discrete transforms based on trigonometric identities and properties of conjugated quadratures of signals.

Optical reference signal 13 is detected by detector 14 to produce an electrical reference signal. The optical reference signal can be derived from a portion of the input beam to the interferometer. Alternatively, the electrical reference signal can be derived directly from the source that introduces the heterodyne frequency splitting in the input beam components (e.g., from the drive signal to an acousto-optical modulator used to generate the heterodyne frequency splitting). The electrical reference signal is converted to a digital format and passed through a high pass filter in converter/filter 54 to produce reference signal S_(R)(t). Reference signal S_(R)(t) in digital format is written as S _(R)(t)=A _(R) cos(φ_(R)+ζ_(R))  (8) where A_(R) and ζ_(R) are the amplitude and offset phase, respectively, of the reference signal. The ADC conversion rate in 54 for S_(R)(t) is the same as the ADC conversion rate in 52 for S(t). The quadrature signal {tilde over (S)}_(R)(t) of S_(R)(t) written as {tilde over (S)} _(R)(t)=A _(R) sin(φ_(R)+ζ_(R))  (9) is generated by electronic processing using measured values of S_(R)(t) according to the formula

$\begin{matrix} {{{\overset{\sim}{S}}_{R}(t)} = {{\left( {\cot\;\omega_{R}\tau} \right){S_{R}\left( {t - {2\tau}} \right)}} - {\frac{\cos\; 2\omega_{R}\tau}{\sin\;\omega_{R}\tau}{S_{R}\left( {t - \tau} \right)}}}} & (10) \end{matrix}$ where 1/τ is the ADC conversion rate of reference signal S_(R)(t) in 54. For the example of a frequency difference for ω_(R)/2π=20 MHz and an ADC conversion rate 1/τ of 120 MHz, Equation (10) reduces to a particularly simple form

$\begin{matrix} {{{\overset{\sim}{S}}_{R}(t)} = {{\frac{1}{\sqrt{3}}\left\lbrack {{S_{R}\left( {t - \tau} \right)} + {S_{R}\left( {t - {2\tau}} \right)}} \right\rbrack}.}} & (11) \end{matrix}$

Reference signal S_(R)(t) and quadrature signal {tilde over (S)}_(R)(t) are conjugated quadratures of reference signal S_(R)(t). The quadrature signal {tilde over (S)}_(R)(t) is generated by processor 16 using Equation (11) or Equation (10) as appropriate.

The quadrature signal {tilde over (S)}(t) of S(t) is generated by processor 56 using the same processing procedure as that described for the generation of quadrature signal {tilde over (S)}_(R)(t). Accordingly, for the example of a ratio of 1/τ and ω_(R)/2π equal to 6,

$\begin{matrix} {\begin{matrix} {{\overset{\sim}{S}(t)} = {\frac{1}{\sqrt{3}}\left\lbrack {{S\left( {t - \tau} \right)} + {S\left( {t - {2\tau}} \right)}} \right\rbrack}} \\ {= {{A_{0}\sin\;\left( {\varphi_{R} + \varphi + \zeta_{1}} \right)} + {{\overset{\sim}{S}}_{\psi}(t)}}} \end{matrix}{where}} & (12) \\ \begin{matrix} {{{\overset{\sim}{S}}_{\psi}(t)} = {{ɛ_{- 1}{\sin\left( {\varphi_{R} - \varphi + \zeta_{- 1}} \right)}} + {ɛ_{0}{\sin\left( {\varphi_{R} + \zeta_{0}} \right)}} +}} \\ {{ɛ_{2}{\sin\left( {\varphi_{R} + {2\varphi} + \zeta_{2}} \right)}} + {ɛ_{3}{{\sin\left( {\varphi_{R} + {3\varphi} + \zeta_{3}} \right)}.}}} \end{matrix} & (13) \end{matrix}$

Equation (12) is valid when the stage slew rate is low, e.g., when φ changes insignificantly over the time period 2τ. Signal S(t) and quadrature signal {tilde over (S)}(t) are conjugated quadratures of signal S(t).

Notably, the integral relationship between 1/τ and ω_(R)/2π allows the generation of feed forward values S′(t) and {tilde over (S)}′(t) of S(t) and {tilde over (S)}(t), respectively, and S_(R)(t) and {tilde over (S)}′_(R)(t) of S_(R)(t) and {tilde over (S)}_(R)(t), respectively, are according to the formulae S′(t)=S(t−6mτ),  (14) {tilde over (S)}′(t)={tilde over (S)}(t−6mτ),  (15) S′ _(R)(t)=S _(R)(t−6mτ),  (16) {tilde over (S)}′ _(R)(t)={tilde over (S)} _(R)(t−6mτ)  (17) where m is an integer such that the error in the phases of feed forward signals with respect to corresponding phases of signals is less than predetermined values set by an end use application. In other words, prior values of the main interference signal and the reference signal can be used to generate the quadrature signals and subsequent error basis functions. In other embodiments in which the ratio between 1/τ and ω_(R)/2π is an integer different from 6, Equations (14)-(17) are modified accordingly.

Using algebraic combinations of the signals S(t), {tilde over (S)}(t), S_(R)(t), and {tilde over (S)}_(R)(t), processing unit 10 generates cyclic error basis functions, which are sine and cosine functions that have the same time-varying arguments as the cyclic error terms, and then uses the cyclic error basis functions to project out respective cyclic error coefficients from S(t) and {tilde over (S)}(t) by low-pass filtering (e.g., averaging). The cyclic error basis functions for S_(ψ0)(t), for example, are especially simple and correspond to the reference signal and its quadrature signal S_(R)(t), and {tilde over (S)}_(R)(t). In other words, to process the signals for information about the cyclic error term ε₀ cos(φ_(R)+ζ₀), signals S_(R)(t) and {tilde over (S)}_(R)(t) are used as time dependent factors in the representation of the cyclic error term ε₀ cos(φ_(R)+ζ₀).

To better understand the representation, it is beneficial to rewrite cyclic error term ε₀ cos(φ_(R)+ζ₀) in terms of the time dependent functions cos(φ_(R)+ζ_(R)) and sin(φ_(R)+ζ_(R)) with the results

$\begin{matrix} {{{ɛ_{0}{\cos\left( {\varphi_{R} + \zeta_{0}} \right)}} = {ɛ_{0}\begin{bmatrix} {{{\cos\left( {\zeta_{0} - \zeta_{R}} \right)}{\cos\left( {\varphi_{R} + \zeta_{R}} \right)}} -} \\ {{\sin\left( {\zeta_{0} - \zeta_{R}} \right)}{\sin\left( {\varphi_{R} + \zeta_{R}} \right)}} \end{bmatrix}}},} & (18) \\ {{ɛ_{0}{\sin\left( {\varphi_{R} + \zeta_{0}} \right)}} = {{ɛ_{0}\begin{bmatrix} {{{\cos\left( {\zeta_{0} - \zeta_{R}} \right)}{\sin\left( {\varphi_{R} + \zeta_{R}} \right)}} +} \\ {{\sin\left( {\zeta_{0} - \zeta_{R}} \right)}{\cos\left( {\varphi_{R} + \zeta_{R}} \right)}} \end{bmatrix}}.}} & (19) \end{matrix}$

Equations (18) and (19) can be rewritten as ε₀ cos(φ_(R)+ζ₀)=[A ₀ cos(φ_(R)+ζ_(R))−B ₀ sin(φ_(R)+ζ_(R))],  (20) ε₀ sin(φ_(R)+ζ₀)=[A ₀ sin(φ_(R)+ζ_(R))+B ₀ cos(φ_(R)+ζ_(R))]  (21) where A ₀=ε₀ cos(ζ₀−ζ_(R)),  (22) B ₀=ε₀ sin(ζ₀−ζ_(R)).  (23)

Conjugated quadratures S(t) and {tilde over (S)}(t) and conjugated quadratures S_(R)(t) and {tilde over (S)}_(R)(t) are transmitted to processor 20 wherein signals Σ₀(t) and {tilde over (Σ)}₀(t) are generated. Signals Σ₀(t) and {tilde over (Σ)}₀(t) are given by the equations Σ₀(t)≡S(t)S _(R)(t)+{tilde over (S)}(t){tilde over (S)} _(R)(t),  (24) {tilde over (Σ)}₀(t)≡−S(t){tilde over (S)} _(R)(t)+{tilde over (S)}(t)S _(R)(t).  (25)

Using properties of conjugated quadratures and certain trigonometric identities, e.g., cos² γ+sin² γ=1, we have

$\begin{matrix} {{{\Sigma_{0}(t)} = {{A_{R}A_{0}} + {A_{R}\begin{bmatrix} {{A_{1}{\cos\left( {\varphi + \zeta_{1} - \zeta_{R}} \right)}} +} \\ {{ɛ_{- 1}{\cos\left( {{- \varphi} + \zeta_{- 1} - \zeta_{R}} \right)}} +} \\ {{ɛ_{2}{\cos\left( {{2\varphi} + \zeta_{2} - \zeta_{R}} \right)}} +} \\ {ɛ_{3}{\cos\left( {{3\varphi} + \zeta_{3} - \zeta_{R}} \right)}} \end{bmatrix}}}},} & (26) \\ {{{\overset{\sim}{\Sigma}}_{0}(t)} = {{A_{R}B_{0}} + {{A_{R}\begin{bmatrix} {{A_{1}{\sin\left( {\varphi + \zeta_{1} - \zeta_{R}} \right)}} +} \\ {{ɛ_{- 1}{\sin\left( {{- \varphi} + \zeta_{- 1} - \zeta_{R}} \right)}} +} \\ {{ɛ_{2}{\sin\left( {{2\varphi} + \zeta_{2} - \zeta_{R}} \right)}} +} \\ {ɛ_{3}{\sin\left( {{3\varphi} + \zeta_{3} - \zeta_{R}} \right)}} \end{bmatrix}}.}}} & (27) \end{matrix}$ Notably, the generation of signals Σ₀(t) and {tilde over (Σ)}₀(t) project the coefficients associated with S_(ψ0)(t) to zero frequency, where low-pass filtering techniques can be used to determine them. Thus, signals Σ₀(t) and {tilde over (Σ)}₀(t) are transmitted to low pass digital filters in processor 24, e.g., low pass Butterworth filters, where coefficients A_(R)A₀ and A_(R)B₀ are determined.

For a Butterworth filter T_(n)(x) of order n, the corresponding outputs of the low pass digital filters for inputs Σ₀(t) and {tilde over (Σ)}₀(t) are

$\begin{matrix} {{{T_{n}\left\lbrack {\Sigma_{0}(t)} \right\rbrack} = {{A_{R}A_{0}} + {A_{R}\begin{bmatrix} {{A_{1}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} + {ɛ_{- 1}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} +} \\ {{ɛ_{2}{O\left( \frac{\omega_{c}}{2\omega_{D}} \right)}^{n}} + {ɛ_{3}{O\left( \frac{\omega_{c}}{3\omega_{D}} \right)}^{n}}} \end{bmatrix}}}},} & (28) \\ \begin{matrix} {{\Sigma_{2}(t)} \equiv {{S_{1} \cdot \Sigma_{1}} - {{\overset{\sim}{S}}_{1} \cdot {\overset{\sim}{\Sigma}}_{1}}}} \\ {= {{A_{R}{A_{1}\begin{bmatrix} {{A_{1}{\cos\left( {\varphi_{R} + {2\varphi} + {2\zeta_{1}} - \zeta_{R}} \right)}} +} \\ {{2ɛ_{- 1}{\cos\left( {\varphi_{R} + \zeta_{1} + \zeta_{- 1} - \zeta_{R}} \right)}} +} \\ {{2ɛ_{2}{\cos\left( {\varphi_{R} + {3\varphi} + \zeta_{1} + \zeta_{2} - \zeta_{R}} \right)}} +} \\ {2ɛ_{3}{\cos\left( {\varphi_{R} + {4\varphi} + \zeta_{1} + \zeta_{3} - \zeta_{R}} \right)}} \end{bmatrix}}} +}} \\ {{A_{R}{O\left( {ɛ_{i}ɛ_{j}} \right)}},} \end{matrix} & (29) \end{matrix}$ where O(x) denotes a term of the order of x, ω_(c) is the −3 dB angular cutoff frequency, and ω_(D)=dφ/dt.

The terms on the right hand sides of Equations (28) and (29) with factors A_(R)A₁ are the sources of the largest errors and accordingly determine the specifications of n and the minimum ratio for ω_(D)/ω_(c) that can be used when the outputs of processor 24 are stored in table 40. For a fourth order Butterworth filter, i.e., n=4, and a minimum ratio for ω_(D)/ω_(c)=7, the error terms on the right hand side of Equations (28) and (29) will generate errors that correspond to ≲0.010 nm (3σ). When the stage is moving at a speed such that the corresponding Doppler shift frequency ω_(D) /2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ω_(D)/ω_(c) is satisfied, the outputs A_(R)A₀ and A_(R)B₀ of the low pass filters in processor 24 are stored in table 40 and in processors 26 and 28 under the control of signal 72 (from processor 70).

Notably, in this preferred embodiment, ω_(D) can vary by factors such as 2 or more during the period associated with output values of A_(R)A₀ and A_(R)B₀ that are stored in table 40.

Quadratures S_(R) and {tilde over (S)}_(R) are transmitted to processor 22 and the value for A_(R) ² is determined in processor 22 by a procedure similar to that used in processor 20 and processor 24 using the formulae T _(n) [S _(R)(t)·S _(R)(t)+{tilde over (S)}_(R)(t)·{tilde over (S)}_(R)(t)]=A _(R) ².  (30)

The value of the order n need only be for example 2. Values of A_(R) ² are transmitted to table 40 and stored under the control of signal 72.

Alternatively, detector 14, converter/filter 54, and processor 16 can be replaced by a lookup table synchronized to the reference signal. This can potentially reduce uncertainty caused by noise on the reference signal. If the value of A_(R) ² is normalized to unity then some equations can be simplified and processor 22 can be removed.

The values for A_(R)A₀, A_(R)B₀, S(t), {tilde over (S)}_(R)(t) S_(R)(t), {tilde over (S)}_(R)(t), and A_(R) ² are transmitted to processor 26 and the values of A_(R)A₀, A_(R)B₀, and A_(R) ² are stored in processor 26 under the control of signal 72 for the generation of conjugated quadratures S₁(t) and {tilde over (S)}₁(t) where

$\begin{matrix} \begin{matrix} {{S_{1}(t)} \equiv {{S(t)} - {\frac{\left( {A_{R}A_{0}} \right)}{A_{R}^{2}}S_{R}} + {\frac{\left( {A_{R}B_{0}} \right)}{A_{R}^{2}}{\overset{\sim}{S}}_{R}}}} \\ {= {{A_{1}\cos\;\left( {\varphi_{R} + \varphi + \zeta_{1}} \right)} + {ɛ_{- 1}{\cos\left( {\varphi_{R} - \varphi + \zeta_{- 1}} \right)}} +}} \\ {{{ɛ_{2}{\cos\left( {\varphi_{R} + {2\varphi} + \zeta_{2}} \right)}} + {ɛ_{3}{\cos\left( {\varphi_{R} + {3\varphi} + \zeta_{3}} \right)}}},} \end{matrix} & (31) \\ \begin{matrix} {{{\overset{\sim}{S}}_{1}(t)} \equiv {{\overset{\sim}{S}(t)} - {\frac{\left( {A_{R}A_{0}} \right)}{A_{R}^{2}}{\overset{\sim}{S}}_{R}} - {\frac{\left( {A_{R}B_{0}} \right)}{A_{R}^{2}}S_{R}}}} \\ {= {{A_{1}\sin\;\left( {\varphi_{R} + \varphi + \zeta_{1}} \right)} + {ɛ_{- 1}{\sin\left( {\varphi_{R} - \varphi + \zeta_{- 1}} \right)}} +}} \\ {{ɛ_{2}{\sin\left( {\varphi_{R} + {2\varphi} + \zeta_{2}} \right)}} + {ɛ_{3}{{\sin\left( {\varphi_{R} + {3\varphi} + \zeta_{3}} \right)}.}}} \end{matrix} & (32) \end{matrix}$

The values for A_(R)A₀, A_(R)B₀, Σ₀(t), and {tilde over (Σ)}₀(t) are transmitted to processor 28 and the values of A_(R)A₀ and A_(R)B₀ are stored in processor 28 under the control of signal 72 for the generation of conjugated quadratures Σ₁(t) and {tilde over (Σ)}₁(t) where

$\begin{matrix} \begin{matrix} {{\Sigma_{1}(t)} \equiv {{\Sigma_{0}(t)} - {A_{R}A_{0}}}} \\ {{= {A_{R}\begin{bmatrix} {{A_{1}{\cos\left( {\varphi + \zeta_{1} - \zeta_{R}} \right)}} + {ɛ_{- 1}{\cos\left( {{- \varphi} + \zeta_{- 1} - \zeta_{R}} \right)}} +} \\ {{ɛ_{2}{\cos\left( {{2\varphi} + \zeta_{2} - \zeta_{R}} \right)}} + {ɛ_{3}{\cos\left( {{3\varphi} + \zeta_{3} - \zeta_{R}} \right)}}} \end{bmatrix}}},} \end{matrix} & (33) \\ \begin{matrix} {{{\overset{\sim}{\Sigma}}_{1}(t)} \equiv {{{\overset{\sim}{\Sigma}}_{0}(t)} - {A_{R}B_{0}}}} \\ {= {{A_{R}\begin{bmatrix} {{A_{1}{\sin\left( {\varphi + \zeta_{1} - \zeta_{R}} \right)}} + {ɛ_{- 1}{\sin\left( {{- \varphi} + \zeta_{- 1} - \zeta_{R}} \right)}} +} \\ {{ɛ_{2}{\sin\left( {{2\varphi} + \zeta_{2} - \zeta_{R}} \right)}} + {ɛ_{3}{\sin\left( {{3\varphi} + \zeta_{3} - \zeta_{R}} \right)}}} \end{bmatrix}}.}} \end{matrix} & (34) \end{matrix}$

Signals S_(R)(t), {tilde over (S)}_(R)(t), Σ₁(t), and {tilde over (Σ)}₁(t) are transmitted to processor 30 for the generation of conjugated quadratures Σ⁻¹(t) and {tilde over (Σ)}⁻¹(t) where

$\begin{matrix} \begin{matrix} {{\Sigma_{- 1}(t)} \equiv {{\Sigma_{1}S_{R}} + {{\overset{\sim}{\Sigma}}_{1}{\overset{\sim}{S}}_{R}}}} \\ {{= {A_{R}^{2}\begin{bmatrix} {{ɛ_{- 1}{\cos\left( {\varphi_{R} + \varphi - \zeta_{- 1} + {2\zeta_{R}}} \right)}} +} \\ {{A_{1}{\cos\left( {\varphi_{R} - \varphi - \zeta_{1} + {2\zeta_{R}}} \right)}} +} \\ {{ɛ_{2}{\cos\left( {\varphi_{R} - {2\varphi} - \zeta_{2} + {2\zeta_{R}}} \right)}} +} \\ {ɛ_{3}{\cos\left( {\varphi_{R} - {3\varphi} - \zeta_{3} + {2\zeta_{R}}} \right)}} \end{bmatrix}}},} \end{matrix} & (35) \\ \begin{matrix} {{{\overset{\sim}{\Sigma}}_{- 1}(t)} \equiv {{\Sigma_{1}{\overset{\sim}{S}}_{R}} - {{\overset{\sim}{\Sigma}}_{1}S_{R}}}} \\ {= {{A_{R}^{2}\begin{bmatrix} {{ɛ_{- 1}{\sin\left( {\varphi_{R} + \varphi - \zeta_{- 1} + {2\zeta_{R}}} \right)}} +} \\ {{A_{1}{\sin\left( {\varphi_{R} - \varphi - \zeta_{1} + {2\zeta_{R}}} \right)}} +} \\ {{ɛ_{2}{\sin\left( {\varphi_{R} - {2\varphi} - \zeta_{2} + {2\zeta_{R}}} \right)}} +} \\ {ɛ_{3}{\sin\left( {\varphi_{R} - {3\varphi} - \zeta_{3} + {2\zeta_{R}}} \right)}} \end{bmatrix}}.}} \end{matrix} & (36) \end{matrix}$

Signals Σ⁻¹(t) and {tilde over (Σ)}⁻¹(t) correspond to the cyclic error basis functions for S_(ψ−1)(t) in that the leading terms of Σ⁻¹(t) and {tilde over (Σ)}⁻¹(t) are sinusoids with the same time-varying argument as that of S_(ψ−1)(t).

Coefficients A_(R) ² A₁A⁻¹ and −A_(R) ² A₁B⁻¹ are next determined through digital low pass filters, e.g., low pass Butterworth filters, in processor 32 where A ⁻¹≡ε⁻¹ cos(ζ⁻¹+ζ₁−2ζ_(R)),  (37) B ⁻¹≡ε⁻¹ sin(ζ⁻¹+ζ₁−2ζ_(R)).  (38)

The input signals for the digital filters are Σ₄(t) and {tilde over (Σ)}₄(t). Input signals Σ₄(t) and {tilde over (Σ)}₄(t) are generated in processor 32 using signals S₁, {tilde over (S)}₁, Σ⁻¹(t), and {tilde over (Σ)}⁻¹(t) according to the formulae Σ₄(t)=[S ₁(t)Σ⁻¹(t)+{tilde over (S)} ₁(t){tilde over (Σ)}⁻¹(t)],  (39) {tilde over (Σ)}₄(t)=[S ₁(t){tilde over (Σ)}⁻¹(t)−{tilde over (S)} ₁(t)Σ⁻¹(t)].  (40)

Equations (39) and (40) are written in terms of A⁻¹ and B⁻¹ using Equations (37) and (38) as

$\begin{matrix} \begin{matrix} {\Sigma_{4} = {{2A_{R}^{2}A_{1}A_{- 1}} +}} \\ {{A_{R}^{2}{A_{1}\begin{bmatrix} {{A_{1}{\cos\left( {{{- 2}\varphi} - \zeta_{1} - \zeta_{1} + {2\zeta_{R}}} \right)}} +} \\ {{2ɛ_{2}{\cos\left( {{{- 3}\varphi} - \zeta_{1} - \zeta_{2} + {2\zeta_{R}}} \right)}} +} \\ {2ɛ_{3}{\cos\left( {{{- 4}\varphi} - \zeta_{1} - \zeta_{3} + {2\zeta_{R}}} \right)}} \end{bmatrix}}} +} \\ {{A_{R}^{2}{O\left( {ɛ_{i}ɛ_{j}} \right)}},} \end{matrix} & (41) \\ \begin{matrix} {{\overset{\sim}{\Sigma}}_{4} = {{{- 2}A_{R}^{2}A_{1}B_{- 1}} +}} \\ {{A_{R}^{2}{A_{1}\begin{bmatrix} {{A_{1}{\sin\left( {{{- 2}\varphi} - \zeta_{1} - \zeta_{1} + {2\zeta_{R}}} \right)}} +} \\ {{2ɛ_{2}{\sin\left( {{{- 3}\varphi} - \zeta_{1} - \zeta_{2} + {2\zeta_{R}}} \right)}} +} \\ {2ɛ_{3}{\sin\left( {{{- 4}\varphi} - \zeta_{1} - \zeta_{3} + {2\zeta_{R}}} \right)}} \end{bmatrix}}} +} \\ {A_{R}^{2}{{O\left( {ɛ_{i}ɛ_{j}} \right)}.}} \end{matrix} & (42) \end{matrix}$

Signals Σ₄(t) and {tilde over (Σ)}₄(t) are sent to low pass digital filters in processor 32, e.g., low pass Butterworth filters, where coefficients 2A_(R) ²A₁A⁻¹ and −2A_(R) ²A₁B⁻¹ are determined. For a Butterworth filter T_(n)(x) of order n, the corresponding outputs of the low pass digital filters for inputs Σ₄(t) and {tilde over (Σ)}₄(t) are

$\begin{matrix} {{{T_{n}\left\lbrack {\Sigma_{4}(t)} \right\rbrack} = {{2A_{R}^{2}A_{1}A_{- 1}} + {A_{R}^{2}{A_{1}\left\lbrack {{A_{1}{O\left( \frac{\omega_{c}}{2\omega_{D}} \right)}^{n}} + {2ɛ_{2}{O\left( \frac{\omega_{c}}{3\omega_{D}} \right)}^{n}} + {2ɛ_{3}{O\left( \frac{\omega_{c}}{4\omega_{D}} \right)}^{n}}} \right\rbrack}}}},} & (43) \\ {{T_{n}\left\lbrack {{\overset{\sim}{\Sigma}}_{4}(t)} \right\rbrack} = {{{- 2}A_{R}^{2}A_{1}B_{- 1}} + {A_{R}^{2}{{A_{1}\left\lbrack {{A_{1}{O\left( \frac{\omega_{c}}{2\omega_{D}} \right)}^{n}} + {2ɛ_{2}{O\left( \frac{\omega_{c}}{3\omega_{D}} \right)}^{n}} + {2ɛ_{3}{O\left( \frac{\omega_{c}}{4\omega_{D}} \right)}^{n}}} \right\rbrack}.}}}} & (44) \end{matrix}$

The terms on the right hand sides of Equations (43) and (44) with factors A_(R) ² A₁ ² are the sources of the largest errors and accordingly determine the specifications of n and the minimum ratio for ω_(D)/ω_(c) that can be used when the outputs of processor 32 are stored in table 40. For a fourth order Butterworth filter, i.e., n=4, and a minimum ratio for ω_(D)/ω_(c)=3.5, the error terms on the right hand side of Equations (43) and (44) will generate errors that correspond to ≲0.010 nm (3σ). The outputs 2A_(R) ² A₁A⁻¹ and −2A_(R) ²A₁B⁻¹ of low pass filters of processor 32 are divided by 2 to generate A_(R) ²A₁A⁻¹ and −A_(R) ²A₁B⁻¹ as the outputs of processor 32. When the stage is moving at a speed such that the corresponding Doppler shift frequency ω_(D)/2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ω_(D)/ω_(c) is satisfied, the outputs A_(R) ²A₁A⁻¹ and −A_(R) ²A₁B⁻¹ of processor 32 are stored in table 40 and in processor 34 under the control of signal 72.

Signals S₁(t) and {tilde over (S)}₁(t) are transmitted to processor 30 for the generation of conjugated quadratures Σ₂(t) and {tilde over (Σ)}₂(t) where

$\begin{matrix} \begin{matrix} {{\Sigma_{2}(t)} \equiv {{S_{1} \cdot \Sigma_{1}} - {{\overset{\sim}{S}}_{1} \cdot {\overset{\sim}{\Sigma}}_{1}}}} \\ {= {{A_{R}{A_{1}\begin{bmatrix} {{A_{1}{\cos\left( {\varphi_{R} + {2\varphi} + {2\zeta_{1}} - \zeta_{R}} \right)}} +} \\ {{2ɛ_{- 1}{\cos\left( {\varphi_{R} + \zeta_{1} + \zeta_{- 1} - \zeta_{R}} \right)}} +} \\ {{2ɛ_{2}{\cos\left( {\varphi_{R} + {3\varphi} + \zeta_{1} + \zeta_{2} - \zeta_{R}} \right)}} +} \\ {2ɛ_{3}{\cos\left( {\varphi_{R} + {4\varphi} + \zeta_{1} + \zeta_{3} - \zeta_{R}} \right)}} \end{bmatrix}}} +}} \\ {{A_{R}{O\left( {ɛ_{i}ɛ_{j}} \right)}},} \end{matrix} & (45) \\ \begin{matrix} {{{\overset{\sim}{\Sigma}}_{2}(t)} \equiv {{S_{1} \cdot {\overset{\sim}{\Sigma}}_{1}} + {{\overset{\sim}{S}}_{1} \cdot \Sigma_{1}}}} \\ {= {{A_{R}{A_{1}\begin{bmatrix} {{A_{1}{\sin\left( {\varphi_{R} + {2\varphi} + {2\zeta_{1}} - \zeta_{R}} \right)}} +} \\ {{2ɛ_{- 1}{\sin\left( {\varphi_{R} + \zeta_{1} + \zeta_{- 1} - \zeta_{R}} \right)}} +} \\ {{2ɛ_{2}{\sin\left( {\varphi_{R} + {3\varphi} + \zeta_{1} + \zeta_{2} - \zeta_{R}} \right)}} +} \\ {2ɛ_{3}{\sin\left( {\varphi_{R} + {4\varphi} + \zeta_{1} + \zeta_{3} - \zeta_{R}} \right)}} \end{bmatrix}}} +}} \\ {A_{R}{{O\left( {ɛ_{i}ɛ_{j}} \right)}.}} \end{matrix} & (46) \end{matrix}$

Signals Σ₂(t) and {tilde over (Σ)}₂(t) correspond to the cyclic error basis functions for S_(ψ2)(t) in that the leading terms of Σ₂(t) and {tilde over (Σ)}₂(t) are sinusoids with the same time-varying argument as that of S_(ψ2)(t).

Signals S₁, {tilde over (S)}₁, Σ₁(t), {tilde over (Σ)}₁(t), Σ₂(t), and {tilde over (Σ)}₂(t) and coefficients A_(R) ²A₁A⁻¹, and are −A_(R) ²A₁B⁻¹ transmitted to processor 34 and coefficients A_(R) ²A₁A⁻¹, and −A_(R) ²A₁B⁻¹ stored in processor 34 under the control of signal 72 for generation of conjugated quadratures Σ₃(t) and {tilde over (Σ)}₃(t) where

$\begin{matrix} \begin{matrix} {\mspace{20mu}{{\Sigma_{3}(t)} \equiv {{\Sigma_{1} \cdot \Sigma_{2}} - {{\overset{\sim}{\Sigma}}_{1} \cdot {\overset{\sim}{\Sigma}}_{2}} + {3\left\lbrack {{A_{R}^{2}A_{1}A_{- 1}S_{1}} - {A_{R}^{2}A_{1}B_{- 1}{\overset{\sim}{S}}_{1}}} \right\rbrack}}}} \\ {= {{A_{R}^{2}{A_{1}^{2}\begin{bmatrix} {{3ɛ_{- 1}{\cos\left( {\varphi_{R} + \varphi + {2\zeta_{1}} + \zeta_{- 1} - {2\zeta_{R}}} \right)}} +} \\ {{A_{1}{\cos\left( {\varphi_{R} + {3\varphi} + {2\zeta_{1}} + \zeta_{1} - {2\zeta_{R}}} \right)}} +} \\ {{3ɛ_{2}{\cos\left( {\varphi_{R} + {4\varphi} + {2\zeta_{1}} + \zeta_{2} - {2\zeta_{R}}} \right)}} +} \\ {3ɛ_{3}{\cos\left( {\varphi_{R} + {5\varphi} + {2\zeta_{1}} + \zeta_{3} - {2\zeta_{R}}} \right)}} \end{bmatrix}}} -}} \\ {{{3\left\lbrack {{A_{R}^{2}A_{1}A_{- 1}S_{1}} - {A_{R}^{2}A_{1}B_{- 1}{\overset{\sim}{S}}_{1}}} \right\rbrack} + {A_{R}^{2}A_{1}{O\left( {ɛ_{i}ɛ_{j}} \right)}} + \ldots}\mspace{11mu},} \end{matrix} & (47) \\ \begin{matrix} {\mspace{20mu}{{\Sigma_{3}(t)} = {{A_{R}^{2}{A_{1}^{2}\begin{bmatrix} {{A_{1}{\cos\left( {\varphi_{R} + {3\varphi} + {3\zeta_{1}} + {2\zeta_{R}}} \right)}} +} \\ {{3ɛ_{2}{\cos\left( {\varphi_{R} + {4\varphi} + {2\zeta_{1}} + \zeta_{2} - {2\zeta_{R}}} \right)}} +} \\ {3ɛ_{3}{\cos\left( {\varphi_{R} + {5\varphi} + {2\zeta_{1}} + \zeta_{3} - {2\zeta_{R}}} \right)}} \end{bmatrix}}} +}}} \\ {{{A_{R}^{2}A_{1}{O\left( {ɛ_{i}ɛ_{j}} \right)}} + \ldots}\mspace{11mu},} \end{matrix} & (48) \\ \begin{matrix} {\mspace{20mu}{{{\overset{\sim}{\Sigma}}_{3}(t)} \equiv {{\Sigma_{1} \cdot {\overset{\sim}{\Sigma}}_{2}} + {{\overset{\sim}{\Sigma}}_{1} \cdot \Sigma_{2}} - {3\left\lbrack {{A_{R}^{2}A_{1}A_{- 1}{\overset{\sim}{S}}_{1}} - {A_{R}^{2}A_{1}B_{- 1}S_{1}}} \right\rbrack}}}} \\ {= {{A_{R}^{2}{A_{1}^{2}\begin{bmatrix} {{3ɛ_{- 1}{\sin\left( {\varphi_{R} + \varphi + {2\zeta_{1}} + \zeta_{- 1} - {2\zeta_{R}}} \right)}} +} \\ {{A_{1}{\sin\left( {\varphi_{R} + {3\varphi} + {2\zeta_{1}} + \zeta_{1} - {2\zeta_{R}}} \right)}} +} \\ {{3ɛ_{2}{\sin\left( {\varphi_{R} + {4\varphi} + {2\zeta_{1}} + \zeta_{2} - {2\zeta_{R}}} \right)}} +} \\ {3ɛ_{3}{\sin\left( {\varphi_{R} + {5\varphi} + {2\zeta_{1}} + \zeta_{3} - {2\zeta_{R}}} \right)}} \end{bmatrix}}} -}} \\ {{3\left\lbrack {{A_{R}^{2}A_{1}A_{- 1}{\overset{\sim}{S}}_{1}} + {A_{R}^{2}A_{1}B_{- 1}S_{1}}} \right\rbrack} + {A_{R}^{2}A_{1}{O\left( {ɛ_{i}ɛ_{j}} \right)}} + \ldots} \end{matrix} & (49) \\ {{{\overset{\sim}{\Sigma}}_{3}(t)} = {{A_{R}^{2}{A_{1}^{2}\begin{bmatrix} {{A_{1}{\sin\left( {\varphi_{R} + {3\varphi} + {3\zeta_{1}} - {2\zeta_{R}}} \right)}} +} \\ {{3ɛ_{2}{\sin\left( {\varphi_{R} + {4\varphi} + {2\zeta_{1}} + \zeta_{2} - {2\zeta_{R}}} \right)}} +} \\ {3ɛ_{3}{\sin\left( {\varphi_{R} + {5\varphi} + {2\zeta_{1}} + \zeta_{3} - {2\zeta_{R}}} \right)}} \end{bmatrix}}} + {A_{R}^{2}A_{1}{O\left( {ɛ_{i}ɛ_{j}} \right)}} + {\ldots\mspace{11mu}.}}} & (50) \end{matrix}$

Signals Σ₃(t) and {tilde over (Σ)}₃(t) correspond to the cyclic error basis functions for S_(ψ3)(t) in that the leading terms of Σ₃(t) and {tilde over (Σ)}₃(t) are sinusoids with the same time-varying argument as that of S_(ψ3)(t).

Coefficients A_(R)A₁ ²A₂ and −A_(R)A₁ ²B₂ (where A₂ and B₂ are cyclic error coefficients for S_(ψ2) and are given explicitly by Equations (67) and (68), respectively, further below) are next determined through digital low pass filters, e.g., low pass Butterworth filters, in processor 38. The input signals for the digital filters are Σ₅(t) and {tilde over (Σ)}₅(t), respectively. The input signals are generated in processor 38 using signals S₁, {tilde over (S)}₁, Σ₂(t), and {tilde over (Σ)}₂(t) according to the formulae Σ₅(t)≡[S ₁(t)Σ₂(t)+{tilde over (S)} ₁(t){tilde over (Σ)}₂(t)],  (51) {tilde over (Σ)}₅(t)≡[S ₁(t){tilde over (Σ)}₂(t)−{tilde over (S)} ₁(t)Σ₂(t)].  (52)

The expansions of Σ₅(t) and {tilde over (Σ)}₅(t), given by Equations (51) and (52), respectively, in terms of cyclic error and non-cyclic error terms are

$\begin{matrix} {{{\Sigma_{5}(t)} = {{A_{R}A_{1}^{2}A_{2}} + {A_{R} A_{1}^{2}\begin{Bmatrix} \begin{matrix} {{ɛ_{- 1}\left\lbrack {{2{\cos\left( {{- \varphi} + \zeta_{- 1} - \zeta_{R}} \right)}} + {\cos\left( {{3\varphi} + {2\zeta_{1}} - \zeta_{- 1} - \zeta_{R}} \right)}} \right\rbrack} +} \\ {{A_{1}{\cos\left( {\varphi + \zeta_{1} - \zeta_{R}} \right)}} + {2ɛ_{2}{\cos\left( {{2\varphi} + \zeta_{2} - \zeta_{R}} \right)}} +} \end{matrix} \\ {ɛ_{3}\left\lbrack {{2{\cos\left( {{3\varphi} + \zeta_{3} - \zeta_{R}} \right)}} + {\cos\left( {{- \varphi} + {2\zeta_{1}} - \zeta_{3} - \zeta_{R}} \right)}} \right\rbrack} \end{Bmatrix}} + {A_{R}A_{1}{O\left( ɛ_{i}^{2} \right)}} + \ldots}}\mspace{11mu},} & (53) \\ {{{{\overset{\sim}{\Sigma}}_{5}(t)} = {{{- A_{R}}A_{1}^{2}B_{2}} + {A_{R}A_{1}^{2}\begin{Bmatrix} {{ɛ_{- 1}\left\lbrack {{2{\sin\left( {{- \varphi} + \zeta_{- 1} - \zeta_{R}} \right)}} + {\sin\left( {{3\varphi} + {2\zeta_{1}} - \zeta_{- 1} - \zeta_{R}} \right)}} \right\rbrack} +} \\ {{A_{1}{\sin\left( {\varphi + \zeta_{1} - \zeta_{R}} \right)}} + {2ɛ_{2}{\sin\left( {{2\varphi} + \zeta_{2} - \zeta_{R}} \right)}} +} \\ {ɛ_{3}\left\lbrack {{2{\sin\left( {{3\varphi} + \zeta_{3} - \zeta_{R}} \right)}} + {\sin\left( {{- \varphi} + {2\zeta_{1}} - \zeta_{3} - \zeta_{R}} \right)}} \right\rbrack} \end{Bmatrix}} + {A_{R}A_{1}{O\left( ɛ_{i}^{2} \right)}} + \ldots}}\mspace{11mu},} & (54) \end{matrix}$ where A₂ and B₂ are given by Equations (67) and (68), respectively, shown further below.

Signals Σ₅(t) and {tilde over (Σ)}₅(t) are sent to low pass digital filters in processor 38, e.g., low pass Butterworth filters, where coefficients A_(R)A₁ ²A₂ and −A_(R)A₁ ²B₂ are determined. For a Butterworth filter T_(n)(x) of order n, the corresponding outputs of the low pass digital filters for inputs Σ₅(t) and {tilde over (Σ)}₅(t) are

$\begin{matrix} {{{T_{n}\left\lbrack {\Sigma_{5}(t)} \right\rbrack} = {{A_{R}A_{1}^{2}A_{2}} + {A_{R}{A_{1}^{2}\begin{bmatrix} {{A_{1}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} + {2ɛ_{- 1}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} + {ɛ_{- 1}{O\left( \frac{\omega_{c}}{3\omega_{D}} \right)}^{n}} +} \\ {{2ɛ_{2}{O\left( \frac{\omega_{c}}{2\omega_{D}} \right)}^{n}} + {2ɛ_{3}{O\left( \frac{\omega_{c}}{3\omega_{D}} \right)}^{n}} + {ɛ_{2}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}}} \end{bmatrix}}}}},} & (55) \\ {\mspace{20mu}{A_{R}^{2}{A_{1}^{4}.}}} & (56) \end{matrix}$

The terms on the right hand sides of Equations (55) and (56) with factors A_(R)A₁ ³ are the sources of the largest errors and accordingly determine the specifications of n and the minimum ratio for ω_(D)/ω_(c) that can be used when the outputs of processor 38 are stored in table 40. For a fourth order Butterworth filter, i.e., n=4, and a minimum ratio for ω_(D)/ω_(c)=7, the error terms on the right hand side of Equations (55) and (56) will generate errors that correspond to ≲0.010 nm (3σ). The outputs A_(R)A₁ ²A₂ and −A_(R)A₁ ²B₂ of low pass filters of processor 38 are the outputs of processor 38. When the stage is moving at a speed such that the corresponding Doppler shift frequency ω_(D) /2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ω_(D)/ω_(c) is satisfied, the outputs A_(R)A₁ ²A₂ and −A_(R)A₁ ²B₂ of processor 38 are stored in table 40 under the control of signal 72.

Coefficients A_(R) ²A₁ ³A₃ and −A_(R) ²A₁ ³B₃ (where A₃ and B₃ are cyclic error coefficients for S_(ψ3) and are given explicitly by Equations (69) and (70), respectively, further below) are next determined through a digital low pass filter, e.g., a low pass Butterworth filter, in processor 36. The input signals for the digital filters are Σ₆(t) and {tilde over (Σ)}₆(t), respectively. The input signals are generated in processor 36 using signals S₁, {tilde over (S)}₁, Σ₃(t), and {tilde over (Σ)}₃(t) according to the formulae Σ₆(t)=[S ₁(t)Σ₃(t)+{tilde over (S)} ₁(t){tilde over (Σ)}₃(t)],  (57) {tilde over (Σ)}₆(t)=[S ₁(t){tilde over (Σ)}₃(t)−{tilde over (S)} ₁(t)Σ₃(t)].  (58)

The expansions of Σ₆(t) and {tilde over (Σ)}₆(t) given by Equations (57) and (58), respectively, in terms of cyclic error and non-cyclic error terms are

$\begin{matrix} {{{\Sigma_{6}(t)} = {{A_{R}^{2}A_{1}^{3}A_{3}} + {A_{R}^{2}{A_{1}^{3}\left\lbrack \begin{matrix} {{ɛ_{- 1}{\cos\left( {{4\varphi} - \zeta_{- 1} + {3\zeta_{1}} - {2\zeta_{R}}} \right)}} +} \\ {{A_{1}{\cos\left( {{2\varphi} - \zeta_{1} + {3\zeta_{1}} - {2\zeta_{R}}} \right)}} +} \\ {{ɛ_{2}{\cos\left( {\varphi - \zeta_{2} + {3\zeta_{1}} - {2\zeta_{R}}} \right)}} + {3ɛ_{2}{\cos\left( {{3\varphi} + \zeta_{1} + \zeta_{2} - {2\zeta_{R}}} \right)}} +} \\ {3ɛ_{3}{\cos\left( {{4\varphi} + \zeta_{1} + \zeta_{3} - {2\zeta_{R}}} \right)}} \end{matrix} \right\rbrack}} + {A_{R}^{2}A_{1}^{2}{O\left( ɛ_{i}^{2} \right)}} + \ldots}}\mspace{11mu},} & (59) \\ {\mspace{20mu}{ɛ_{- 1}.}} & (60) \end{matrix}$ where A₃ and B₃ are given by Equations (69) and (70), respectively.

Signals Σ₆(t) and {tilde over (Σ)}₆(t) are sent to low pass digital filters in processor 36, e.g., low pass Butterworth filters, where coefficients A_(R) ² A₁ ³ A₃ and −A_(R) ² A₁ ³B₃ are determined. For a Butterworth filter T_(n)(x) of order n, the corresponding outputs of the low pass digital filters for inputs Σ₆(t) and {tilde over (Σ)}₆(t) are

$\begin{matrix} {{{T_{n}\left\lbrack {\Sigma_{6}(t)} \right\rbrack} = {{A_{R}^{2}A_{1}^{3}A_{3}} + {A_{R}^{2}{A_{1}^{3}\begin{bmatrix} {{A_{1}{O\left( \frac{\omega_{c}}{2\omega_{D}} \right)}^{n}} + {ɛ_{- 1}{O\left( \frac{\omega_{c}}{4\omega_{D}} \right)}^{n}} +} \\ {{ɛ_{2}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} + {3ɛ_{2}{O\left( \frac{\omega_{c}}{3\omega_{D}} \right)}^{n}} + {3ɛ_{3}{O\left( \frac{\omega_{c}}{4\omega_{D}} \right)}^{n}}} \end{bmatrix}}}}},} & (61) \\ {{T_{n}\left\lbrack {\Sigma_{6}(t)} \right\rbrack} = {{{- A_{R}^{2}}A_{1}^{3}B_{3}} + {A_{R}^{2}{{A_{1}^{3}\begin{bmatrix} {{A_{1}{O\left( \frac{\omega_{c}}{2\omega_{D}} \right)}^{n}} + {ɛ_{- 1}{O\left( \frac{\omega_{c}}{4\omega_{D}} \right)}^{n}} +} \\ {{ɛ_{2}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} + {3ɛ_{2}{O\left( \frac{\omega_{c}}{3\omega_{D}} \right)}^{n}} + {3ɛ_{3}{O\left( \frac{\omega_{c}}{4\omega_{D}} \right)}^{n}}} \end{bmatrix}}.}}}} & (62) \end{matrix}$

The terms on the right hand sides of Equations (61) and (62) with factors A_(R) ² A₁ ⁴ are the sources of the largest errors and accordingly determined the specifications of n and the minimum ratio for ω_(D)/ω_(c) that can be used when the outputs of processor 36 are stored in table 40. For a fourth order Butterworth filter, i.e., n=4, and a minimum ratio for ω_(D)/(c=3.5, the error terms on the right hand side of Equations (61) and (62) will generate errors that correspond to ≲0.010 nm (3σ). The outputs A_(R) ² A₁ ³A₃ and −A_(R) ²A₁ ³B₃ of low pass filters of processor 36 are the outputs of processor 36. When the stage is moving at a speed such that the corresponding Doppler shift frequency ω_(D) /2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ω_(D)/ω_(c) is satisfied, the outputs A_(R) ²A₁ ³A₃ and −A_(R) ²A₁ ³B₃ of processor 36 are stored in table 40 under the control of signal 72.

Finally, quadratures S and {tilde over (S)} are transmitted from processors 52 and 56 respectively to processor 18 for the purpose of determining a value for A₁ ². First, a signal S(t)S(t)+{tilde over (S)}(t){tilde over (S)}(t) is generated where

$\begin{matrix} {{{{S(t)}{S(t)}} + {{\overset{\sim}{S}(t)}{\overset{\sim}{S}(t)}}} = {A_{1}^{2} + \left\lbrack {ɛ_{- 1}^{2} + ɛ_{0}^{2} + ɛ_{2}^{2} + ɛ_{3}^{2}} \right\rbrack + {2A_{1}ɛ_{- 1}{\cos\left( {{2\varphi} + \zeta_{1} - \zeta_{- 1}} \right)}} + {2A_{1}ɛ_{0}{\cos\left( {\varphi + \zeta_{1} - \zeta_{0}} \right)}} + {2A_{1}ɛ_{2}{\cos\left( {{- \varphi} + \zeta_{1} - \zeta_{2}} \right)}} + {2A_{1}ɛ_{3}{\cos\left( {{{- 2}\varphi} + \zeta_{1} - \zeta_{3}} \right)}} + {{O\left( {ɛ_{i}ɛ_{j}} \right)}.}}} & (63) \end{matrix}$

The signal of Equation (63) is sent to a low pass digital filter in processor 18, e.g., a low pass Butterworth filter, where the coefficient A₁ ² is determined. For a Butterworth filter T_(n)(x) of order n, the corresponding outputs of the low pass digital filter for the signal of Equation (63) is

$\begin{matrix} {{T_{n}\left\lbrack {{{S(t)} \cdot {S(t)}} + {{\overset{\sim}{S}(t)} \cdot {\overset{\sim}{S}(t)}}} \right\rbrack} = {A_{1}^{2} + \left\lbrack {ɛ_{- 1}^{2} + ɛ_{0}^{2} + ɛ_{2}^{2} + ɛ_{3}^{2}} \right\rbrack + {{A_{1}\begin{bmatrix} {{2ɛ_{- 1}{O\left( \frac{\omega_{c}}{2\omega_{D}} \right)}^{n}} + {2ɛ_{0}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} +} \\ {{2ɛ_{2}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} + {2ɛ_{3}{O\left( \frac{\omega_{c}}{2\omega_{D\;}} \right)}^{n}}} \end{bmatrix}}.}}} & (64) \end{matrix}$

The accuracy required for the determination of A₁ ² is approximately 0.5% in order to limit errors generated in the computation of cyclic error signals S_(ψj) to ≲0.010 nm (3σ). Therefore the error terms ε⁻¹ ², ε₀ ², ε₂ ², and ε₂ ² on the right hand side of Equation (64) are negligible. The terms on the right hand side of Equation (64) of the form

${O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}$ are the sources of the largest Doppler shift frequency dependent errors and accordingly determine the specifications of n and the minimum ratio for ω_(D)/ω_(c) that can be used when the output of processor 18 is stored in table 40. For a second order Butterworth filter, i.e., n=2, and a minimum ratio for ω_(D)/ω_(c)=3.5, the Doppler shift frequency dependent error terms on the right hand side of Equation (64) will generate errors that correspond to ≲0.010 nm (3σ). The output A₁ ² of the low pass filter of processor 18 is the output of processor 18. When the stage is moving at a speed such that the corresponding Doppler shift frequency ω_(D)/2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ω_(D)/ω_(c) is satisfied, the output A₁ ² of processor 18 is stored in table 40 under the control of signal 72.

Referring now to FIG. 4 b, processor 60 generates the compensating error signal S_(ψ). With respect to generating signal S_(ψ), it is beneficial to rewrite the ε⁻¹, ε₀, ε₂, and ε₃ cyclic error terms of S_(ψ) in terms of the highest order time dependent terms of Σ⁻¹(t), {tilde over (Σ)}⁻¹(t), S_(R), {tilde over (S)}_(R), Σ₂(t), {tilde over (Σ)}₂(t), Σ₃(t), and {tilde over (Σ)}₃(t), i.e., cos(φ_(R)−φ−ζ₁+2ζ_(R)), sin(φ_(R)−φ−ζ₁+2ζ_(R)), cos(φ_(R)+ζ_(R)), sin(φ_(R)+ζ_(R)), cos(φ_(R)+2φ+2ζ₁−ζ_(R)), sin(φ_(R)+2φ−2ζ₁+2ζ₁−ζ_(R)), cos(φ_(R)+3φ+3ζ₁−2ζ_(R)), and sin(φ_(R)+3φ+3ζ₁−2ζ_(R)) as

$\begin{matrix} {{S_{\psi}( t)} = {\left\lbrack \begin{matrix} {ɛ_{- 1}{\cos\left( {\zeta_{1} + \zeta_{- 1} - {2\zeta_{R}}} \right)}{\cos\left( {\varphi_{R} - \varphi - \zeta_{1} + {2\zeta_{R}}} \right)}} \\ {{- ɛ_{- 1}}{\sin\left( {\zeta_{1} + \zeta_{- 1} - {2\zeta_{R}}} \right)}{\sin\left( {\varphi_{R} - \varphi - \zeta_{1} + {2\zeta_{R}}} \right)}} \end{matrix} \right\rbrack + \mspace{59mu}\left\lbrack \begin{matrix} {ɛ_{0}{\cos\left( {\zeta_{0} - \zeta_{R}} \right)}{\cos\left( {\varphi_{R} + \zeta_{R}} \right)}} \\ {{- ɛ_{0}}{\sin\left( {\zeta_{0} - \zeta_{R}} \right)}{\sin\left( {\varphi_{R} + \zeta_{R}} \right)}} \end{matrix} \right\rbrack + \mspace{40mu}\left\lbrack \begin{matrix} {ɛ_{2}{\cos\left( {{{- 2}\zeta_{1}} + \zeta_{2} + \zeta_{R}} \right)}{\cos\left( {\varphi_{R} + {2\varphi} + {2\zeta_{1}} - \zeta_{R}} \right)}} \\ {{- ɛ_{2}}{\sin\left( {{{- 2}\zeta_{1}} + \zeta_{2} + \zeta_{R}} \right)}{\sin\left( {\varphi_{R} + {2\varphi} + {2\zeta_{1}} - \zeta_{R}} \right)}} \end{matrix} \right\rbrack + \mspace{56mu}{\left\lbrack \begin{matrix} {ɛ_{3}{\cos\left( {{{- 3}\zeta_{1}} + \zeta_{3} + {2\zeta_{R}}} \right)}{\cos\left( {\varphi_{R} + {3\varphi} + {3\zeta_{1}} - {2\zeta_{R}}} \right)}} \\ {{- ɛ_{3}}{\sin\left( {{{- 3}\zeta_{1}} + \zeta_{3} + {2\zeta_{R}}} \right)}{\sin\left( {\varphi_{R} + {3\varphi} + {3\zeta_{1}} - {2\zeta_{R}}} \right)}} \end{matrix} \right\rbrack.}}} & (65) \end{matrix}$ Equation (65) for S_(ψ) is next written in the form

$\begin{matrix} {{S_{\psi}(t)} = {\left\lbrack {{A_{- 1}{\cos\left( {\varphi_{R} - \varphi - \zeta_{1} + {2\zeta_{R}}} \right)}} - {B_{- 1}{\sin\left( {\varphi_{R} - \varphi - \zeta_{1} + {2\zeta_{R}}} \right)}}} \right\rbrack + \left\lbrack {{A_{0}{\cos\left( {\varphi_{R} + \zeta_{R}} \right)}} - {B_{0}{\sin\left( {\varphi_{R} + \zeta_{R}} \right)}}} \right\rbrack + \left\lbrack {{A_{2}{\cos\left( {\varphi_{R} + {2\varphi} + {2\zeta_{1}} - \zeta_{R}} \right)}} - {B_{2}{\sin\left( {\varphi_{R} + {2\varphi} + {2\zeta_{1}} - \zeta_{R}} \right)}}} \right\rbrack + \left\lbrack {{A_{3}{\cos\left( {\varphi_{R} + {3\varphi} + {3\zeta_{1}} - {2\zeta_{R}}} \right)}} - {B_{3}{\sin\left( {\varphi_{R} + {3\varphi} + {3\zeta_{1}} - {2\zeta_{R}}} \right)}}} \right\rbrack}} & (66) \end{matrix}$ where A⁻¹, B⁻¹, A₀, and B₀ are given by equations (37), (38), (22), and (23), respectively, and A ₂=ε₂ cos(−2ζ₁+ζ₂+ζ_(R)),  (67) B ₂=ε₂ sin(−2ζ₁+ζ₂+ζ_(R)),  (68) A ₃=ε₃ cos(−3ζ₁+ζ₃+2ζ_(R)),  (69) B ₃=ε₃ sin(−3ζ₁+ζ₃+2ζ_(R)).  (70)

Compensation error signal S_(ψ) is generated in processor 44 using Equation (66), the coefficients transmitted from table 40 as signal 42, and the signals Σ⁻¹(t), {tilde over (Σ)}⁻¹(t), S_(R), {tilde over (S)}_(R), Σ₂(t), {tilde over (Σ)}₂(t), Σ₃(t), and {tilde over (Σ)}₃(t) (which comprise the cyclic error basis functions) under control of control of signal 74 (from processor 70). Explicitly,

$\begin{matrix} {{{S_{\psi}(t)} = {\left\lbrack {{\left( \frac{A_{R}^{2}A_{1}A_{- 1}}{\left( A_{R}^{2} \right)^{2}A_{1}^{2}} \right)\Sigma_{- 1}} + {\left( \frac{{- A_{R}^{2}}A_{1}B_{- 1}}{\left( A_{R}^{2} \right)^{2}A_{1}^{2}} \right){\overset{\sim}{\Sigma}}_{- 1}}} \right\rbrack + \left\lbrack {{\left( \frac{A_{R}A_{0}}{A_{R}^{2}} \right)S_{R}} - {\left( \frac{A_{R}B_{0}}{A_{R}^{2}} \right){\overset{\sim}{S}}_{R}}} \right\rbrack + \left\lbrack {{\left( \frac{A_{R}A_{1}^{2}A_{2}}{{A_{R}^{2}\left( A_{1}^{2} \right)}^{2}} \right)\Sigma_{2}} + {\left( \frac{{- A_{R}}A_{1}^{2}B_{2}}{{A_{R}^{2}\left( A_{1}^{2} \right)}^{2}} \right){\overset{\sim}{\Sigma}}_{2}}} \right\rbrack + \left\lbrack {{\left( \frac{A_{R}^{2}A_{1}^{3}A_{3}}{\left( A_{R}^{2} \right)^{2}\left( A_{1}^{2} \right)^{3}} \right)\Sigma_{3}} + {\left( \frac{{- A_{R}^{2}}A_{1}^{3}B_{3}}{\left( A_{R}^{2} \right)^{2}\left( A_{1}^{2} \right)^{3}} \right){\overset{\sim}{\Sigma}}_{3}}} \right\rbrack}}} & (71) \end{matrix}$ In other words, the compensation error signal is generated form a superposition of the error basis functions weighted by the cyclic error coefficients.

The compensating signal S_(ψ) is subtracted from signal S in processor 46 under control of signal 76 (from processor 70) to generate compensated signal S−S_(ψ). Control signal 76 determines when signal S is to be compensated. The phase φ=4kL is then extracted from the compensated signal with a subsequent processor (not shown) to, for example, provide a more accurate measurement of the distance L.

In the presently preferred embodiment, the error compensation signal S_(ψ)(t) is subtracted from prior values of the signals S(t). For example, feedforward signal S′(t), (as described in Equation (14)) may replace signal S(t). The delay, m, of the feedforward signal is chosen to be equal to the processing delay in calculating the error basis functions and subsequently S_(ψ)(t) from signal S(t). In this manner, S′(t) and S_(ψ)(t) represent the same time of input signal S(t).

In further embodiments, the cyclic error coefficients may be stored and updated at a lower data rate than that used to generate the cyclic error basis functions from the feedforward values. In such cases, the stored values for the cyclic error coefficients may used for the calculation of the cyclic error basis functions as necessary. Of course, in yet further embodiments, the coefficients and/or the error basis functions can be calculated in real time, without the use of the feedforward signals.

An important assumption in the preferred embodiment of the cyclic error compensation described above is that the Doppler shift frequency dφ(t)/dt is small enough relative to the heterodyne frequency ω_(R), that the quadrature signal {tilde over (S)}(t) can be approximated (in analogy to Equation (10) by the expression:

$\begin{matrix} {{\overset{\sim}{S}(t)} = {{\left( {\cot\;\omega_{R}\tau} \right){S\left( {t - {2\tau}} \right)}} - {\frac{\cos\; 2\;\omega_{R}\tau}{\sin\;\omega_{R}\tau}{S\left( {t - \tau} \right)}}}} & (72) \end{matrix}$ or the simpler expression given by Equation (12). In further embodiments, the quadrature signal {tilde over (S)}(t) may be more accurately calculated according to:

$\begin{matrix} {{\overset{\sim}{S}(t)} = {{\left( {\cot\;\omega_{M}\tau} \right){S\left( {t - {2\tau}} \right)}} - {\frac{\cos\; 2\;\omega_{M}\tau}{\sin\;\omega_{M}\tau}{S\left( {t - \tau} \right)}}}} & (73) \end{matrix}$ where ω_(M) is the instantaneous rate of change of the phase of the main interference signal S(t), which can be determined with sufficient accuracy by assuming that the cyclic error contributions to S(t) are negligible.

Also, in further embodiments, the cyclic error compensation technique may be used for cyclic error terms different from those explicitly described in Equations (3)-(6). In particular, using algebraic combinations of the signals S(t), {tilde over (S)}(t), S_(R)(t), and {tilde over (S)}_(R)(t), a processing unit can generate cyclic error basis functions, which are sine and cosine functions that have the same time-varying arguments as the cyclic error terms that need to be compensated, and then use the cyclic error basis functions to project out respective cyclic error coefficients from S(t) and {tilde over (S)}(t) by low-pas filtering (e.g., averaging).

For example, to determine the coefficients for a half-cycle cyclic error of the form:

$\begin{matrix} {S_{\psi{({1/2})}} = {ɛ_{1/2}{\cos\left( {\varphi_{R} + \frac{\varphi}{2} + \zeta_{1/2}} \right)}}} & (74) \end{matrix}$ one can calculate cyclic error basis functions for the half-cycle cyclic error as follows. First calculate signals Σ′₇(t) and {tilde over (Σ)}′₇(t) as:

$\begin{matrix} {{{\Sigma_{7}^{\prime}(t)} = \sqrt{\frac{\Sigma_{0} - \left( {{A_{R}A_{0}} - {A_{R}A_{1}}} \right)}{2A_{R}A_{1}}}},} & (75) \\ {{{\overset{\sim}{\Sigma}}_{7}^{\prime}(t)} = {\sqrt{\frac{\left( {{A_{R}A_{0}} + {A_{R}A_{1}}} \right) - \Sigma_{0}}{2A_{R}A_{1}}}.}} & (76) \end{matrix}$ Notably, the leading term of Σ′₇(t) is |cos(φ/2+ζ₁/2−ζ_(R)/2)|, and the leading term of {tilde over (Σ)}′₇(t) is |sin(φ/2+ζ₁/2−ζ_(R)/2)|. Zero phase crossings in Σ′₇(t) and {tilde over (Σ)}′₇(t) are then measured to remove the absolute value operation and define Σ₇(t) and {tilde over (Σ)}₇(t), which have leading terms cos(φ/2+ζ₁/2−ζ_(R)/2) and sin(φ/2+ζ₁/2−ζ_(R)/2), respectively.

Half-cycle error basis functions Σ_(1/2)(t) and {tilde over (Σ)}_(1/2)(t) are then generated as: Σ_(1/2)(t)=Σ₇(t)S _(R)(t)−{tilde over (Σ)}₇(t){tilde over (S)} _(R)(t),  (77) {tilde over (Σ)}_(1/2)(t)={tilde over (Σ)}₇(t)S _(R)(t)+Σ₇(t){tilde over (S)} _(R)(t),  (78) where Σ_(1/2) (t) and {tilde over (Σ)}_(1/2) (t) have as their leading terms sine and cosine functions, respectively, whose time-dependent argument is the same as that of S_(ψ(1/2)). To determine the coefficients for S_(ψ(1/2)), one projects half-cycle error basis functions onto S(t) and {tilde over (S)}(t) to move the half-cycle cyclic error component to zero frequency, for example: Σ₈(t)=S(t)Σ_(1/2)(t)+{tilde over (S)}(t){tilde over (Σ)}_(1/2)(t),  (79) {tilde over (Σ)}₈(t)={tilde over (S)}(t)Σ_(1/2)(t)−S(t){tilde over (Σ)}_(1/2)(t).  (80). Low-pass filtering (e.g., with the Butterworth filter) of Σ₈(t) and {tilde over (Σ)}₈(t) then yield the half-cycle error coefficients in analogy to the extraction of the previously described cyclic error coefficients. In particular, the leading terms following the low-pass filtering are A_(R)A_(1/2) cos(ζ_(1/2)−ζ₁/2−ζ_(R)/2) and A_(R)A_(1/2) sin(ζ_(1/2)−ζ₁/2−ζ_(R)/2), respectively.

We now describe the CMS approach for the CEC, in which a cyclic error correction signal D_(ψ)(t) is subtracted from a corresponding DFT signal D(t) of the electrical interference signal S(t) to produce a compensated DFT signal. FIG. 4 c shows a simplified schematic diagram of a measurement using the CMS approach. The optical interference signal 111 is received and amplified by photoelectric receiver 112. The resulting electrical interference signal 113 is filtered by lowpass filter (LPF) 114 producing filtered signal 115. The LPF 114 is designed to prevent harmonics of the interference signal 111 from being aliased into the frequency range of interest. Filtered signal 115 is digitized by ADC 116, to produce digitized measurement signal 117. A typical ADC for a high performance displacement measuring interferometer may have 12 bits of resolution at sampling rates of 120 MHz. The digitized measurement signal 117 is processed by phase meter 120 (described below) to produce outputs magnitude 125 and phase 127 which represent the digitized measurement signal 117 as a transform. The magnitude output 125 is used for status and diagnostic purposes. The phase output 127 is used by position calculator 130 which is fully described in published U.S. application Ser. No. 10/211,435 (publication number US 2003/0025914 A1), incorporated herein by reference. Position calculator 130 calculates measured position 131 and estimated speed 133. The measured position 131 is filtered by digital filter 136, which is fully described in U.S. Pat. No. 5,767,972, incorporated herein by reference, to generate filtered position signal 137. Filtered position signal 137 represents the desired measurement of the distance L.

Phase meter 120 includes a Discrete Fourier Transform (DFT) processor 122, a cyclic error compensation (CEC) calculator 140, and a Coordinate Rotation by Digital Computer (CORDIC) converter 124. Signals 123, 143, 145, and 147 are complex values, which consist of both a real component and an imaginary component, as a+jb, where a is the real component, b is the imaginary component, and j is √{square root over (−1)}. (The symbol i is sometimes used in the literature instead of j.) Other representations of complex or quadrature values can be used, and may be expressed using other symbols such as, for example, I and Q, or X and Y, or A and Ã. Complex values may be converted from rectangular (real and imaginary) representation to polar (magnitude and phase angle) representation. The numeric representation of the digital signals may be integer, fractional, or floating point.

The DFT processor 122 converts a series of consecutive samples of digitized measurement signal 117 into a complex measurement signal 123 representing a transform of the digitized measurement signal 117 at a selected center frequency of DFT processor 122. The center frequency is determined by control circuitry (not shown) and the estimated speed 133 is determined by position calculator 130.

An exemplary DFT processor 122 is a 72-point windowed DFT performed at a 10 MHz rate. Since the DFT calculation is being updated in time, the complex measurement signal 123 is expressed as a function of time D(t). This 10 MHz update rate provides 83% overlap of samples of the digitized measurement signal 117 between one DFT calculation and the next for an ADC sampling rate of f_(S)=120 MHz. A typical window function is the Blackman window, which reduces errors due to the discontinuities at the beginning and end of the series of digitized measurement signal samples used for the DFT.

The CEC calculator 140 calculates and compensates for certain of the cyclic errors. CEC error estimator 144 (described in more detail below with reference to FIG. 4 d) calculates complex error compensation signal 145. Optional delay 142, and other delays (not shown) in CEC calculator 140 may be used to match the processing delay of the various calculations. Adder 146 combines delayed complex measurement signal 143 with complex error compensation signal 145 to produce compensated complex measurement signal 147, in which the certain cyclic error signals are substantially reduced.

CORDIC converter 124 converts the compensated complex measurement signal 147 to magnitude 125 and phase 127.

Referring to FIG. 4 d, the CEC error estimator 144 includes two processing units. One processing unit 148 determines error basis functions and complex factors relating to the amplitudes and offset phases of the certain cyclic errors that need be compensated. A second processing unit 204 generates complex error compensation signal D_(ψ)(t) 145 using the error basis functions and complex factors relating to the amplitudes and offset phases determined by first processing unit 148.

The first processing unit 148 for one embodiment is shown schematically in FIG. 4 d and the second processing unit 204 of this embodiment is shown schematically in FIG. 4 e. These processing units are incorporated into the architecture shown in FIG. 4 c that may also include any of a variety of other techniques such as a glitch filter (as described in published U.S. application Ser. No. 10/211,435), dynamic data age adjustment (as described in U.S. Pat. No. 6,597,459, incorporated herein by reference), and digital filtering as described in U.S. Pat. No. 5,767,972.

Note that, as in the DMS approach, the set of cyclic error terms whose phases are independent of φ_(R) has been omitted from Equation (2) because, in this CMS approach, they are eliminated by LPF (Lowpass filter) 114 and the bandpass filtering inherent in DFT processor 122.

Note that there are conditions where the cyclic error terms 82 or 83 may be aliased and appear at or near the frequency of the measurement signal. Let f_(R) be the reference frequency (f_(R)=(O_(R)/2π), f_(D) be the Doppler shift frequency (f_(D)=ω_(D) /2π), and f_(M) be the resulting measurement frequency. When f_(D)=−(⅔)f_(R),

$\begin{matrix} {{f_{M} = {{f_{R} + f_{D}} = {{f_{R} - {\frac{2}{3}f_{R}}} = {\frac{1}{3}f_{R}}}}},{f_{ɛ_{2}} = {{f_{R} + {2f_{D}}} = {{f_{R} - {\frac{4}{3}f_{R}}} = {{- \frac{1}{3}}{f_{R}.{Also}}}}}},{{{when}\mspace{14mu} f_{D}} = {{- \left( {1/2} \right)}f_{R}}},} & (81) \\ {{f_{M} = {{f_{R} + f_{D}} = {{f_{R} - {\frac{1}{2}f_{R}}} = {\frac{1}{2}f_{R}}}}},{f_{ɛ_{3}} = {{f_{R} + {3f_{D}}} = {{f_{R} - {\frac{3}{2}f_{R}}} = {{- \frac{1}{2}}{f_{R}.}}}}}} & (82) \end{matrix}$ In both cases, the negative frequency is indistinguishable from the corresponding positive frequency, and measurement accuracy is affected.

Complex factors relating to amplitudes ε_(p) and offset phases ζ_(p) of the four cyclic error terms and time dependent factors of the cyclic error terms are generated using processed values D(t) 123 from DFT processor 122. The factors are stored in registers 162, 176, 186, and 192 for subsequent use in generation of the cyclic error correction signal D_(ψ)(t) 145. The time dependent factors of the cyclic error terms are obtained by application of digital transforms based on trigonometric identities and properties of complex signals.

DFT processor 122 calculates the complex DFT of the digitized measurement signal 117 as:

$\begin{matrix} {{{D_{q}\left( t_{1} \right)} = {\sum\limits_{n = 0}^{N - 1}{W_{n}{S\left( {{n\;\tau} + t_{1}} \right)}\;\exp\;\left\{ {- {j\left( {2{\pi\left( {n - \frac{N - 1}{2}} \right)}{q/N}} \right)}} \right\}}}},} & (83) \end{matrix}$ for 0<q<N−1, where τ=1/f_(S), W_(n) is a window function centered at

${n = \frac{N - 1}{2}},$ and t₁ is the time at which the DFT calculation is updated. When q is selected by control circuitry (not shown) as an integer approximately equal to Nf_(M)/f_(S), to correspond to the center frequency of the primary component of the digitized measurement signal. A typical value for N is 72, and a typical window function W_(n) is the Blackman window function.

The equation for the DFT is arranged so that the center of the sequence of the N consecutive samples of the digitized measurement signal 117 (n=35.5 in this case) corresponds to zero phase and the center of the window function. Therefore, since the window function and DFT coefficients (the complex exponential or equivalently the “cos+j sin” factors shown below) are symmetrical about the center, and window weighting is zero at the ends, a change in the value of q, has a reduced effect on the phase of the result.

The transform signal D_(q)(t₁) is updated at a rate f_(U) that is lower than the rate f_(S) at which the signal S(t) is sampled. In this example, t₁=lΔt₁ (where l is an integer and Δt₁≡1/f_(U) is the update interval) and f_(U)=10 MHz.

Alternatively, the DFT equation can be “folded” to reduce the number of multiplication operations that are performed and calculated as:

$\begin{matrix} {{D_{q}\left( t_{1} \right)} = {\sum\limits_{n = 0}^{\frac{N}{2} - 1}{{W_{n}\left( {{x_{n}\left( t_{1} \right)} + {x_{N - 1 - n}\left( t_{1} \right)}} \right)}\begin{pmatrix} {{\cos\left( {2\pi\frac{n + 0.5}{N}\frac{q}{8}} \right)} +} \\ {j\;\sin\;\left( {2\pi\frac{n + 0.5}{N}\frac{q}{8}} \right)} \end{pmatrix}}}} & (84) \end{matrix}$ for 0<q<8N−1, where x_(n)(t₁) for n=0, . . . , N−1 are N consecutive samples (i.e., t=nτ) of S(t+t₁), and W_(n) is centered at n=0. In this example, the larger range of q yields a more finely spaced resolution of ⅛ bin to reduce amplitude variations (or “picket fence” effect) as the frequency changes from one bin to the next.

The DFT function is equivalent to a mixing and a filtering operation. The mixing is a result of multiplying the input data by the complex exponential or equivalently the “cos+j sin” factor. The filtering is a result of the summation and the window function W_(n).

Since the window function W_(n) makes terms zero outside the range of summation, the DFT expression can be written as a sum over all n. In a simplified expression for an “unfolded” DFT (as in equation (83)), the digitized measurement signal S(nτ+t₁) 177 has been approximated by its primary term A₁ cos(φ_(R)(nτ+t₁)+φ(nτ+t₁)+ζ₁), where explicit sampling (t=nτ) has been included:

$\begin{matrix} {{D\left( t_{1} \right)} = {A_{1}{\sum\limits_{n}{W_{n}{\cos\begin{pmatrix} {{\varphi_{R}\left( {{n\;\tau} + t_{1}} \right)} +} \\ {{\varphi\left( {{n\;\tau} + t_{1}} \right)} + \zeta_{1}} \end{pmatrix}}\begin{pmatrix} {{\cos\left( {\varphi_{C}\left( {n,q} \right)} \right)} +} \\ {{jsin}\left( {\phi_{C}\left( {n,q} \right)} \right)} \end{pmatrix}}}}} & (85) \end{matrix}$ where φ_(R)(nτ+t₁)=ω_(R)nτ+ω_(R)t₁ and φ(nτ+t₁)=ω_(D)nτ+ω_(D)t₁+φ₀ is the sampled version of the Doppler phase shift φ(t)=ω_(D)t+φ₀, and

${\varphi_{C}\left( {n,q} \right)} = {2\pi\;\frac{nq}{N}}$ is the phase of the DFT with q≈Nf_(M)/f_(S), yielding φ_(C)(n,q)≈2πf_(M)nτ=ω_(M)nτ. (The q in D_(q)(t₁) has been suppressed so that D(t₁)≡D_(q)(t₁).) Using trigonometric identities, equation (85) can be expanded to (with time dependent arguments temporarily suppressed):

$\begin{matrix} {{D\left( t_{1} \right)} = {A_{1}{W_{n}\begin{bmatrix} {{\frac{1}{2}\begin{pmatrix} {{\cos\left( {\varphi_{R} + \varphi + \zeta_{1} + \varphi_{C}} \right)} +} \\ {\cos\left( {\varphi_{R} + \varphi + \zeta_{1} - \varphi_{C}} \right)} \end{pmatrix}} +} \\ {j\frac{1}{2}\begin{pmatrix} {{\sin\left( {\varphi_{R} + \varphi + \zeta_{1} + \varphi_{C}} \right)} -} \\ {\sin\left( {\varphi_{R} + \varphi + \zeta_{1} - \varphi_{C}} \right)} \end{pmatrix}} \end{bmatrix}}}} & (86) \end{matrix}$ The terms containing φ_(R)+φ+ζ₁+φ_(C) are high frequency sinusoids varying with n that are filtered out in the summation including the window function W_(n) that covers many cycles. The constant ½ may be dropped for convenience. The terms containing (φ_(R)+φ+ζ₁−φ_(C) that are slowly varying in the summation over the window remain:

$\begin{matrix} {{D\left( t_{1} \right)} \approx {A_{1}{\sum\limits_{n}{W_{n}\begin{bmatrix} {{\cos\begin{pmatrix} {{\omega_{R}n\;\tau} + {\omega_{R}t_{1}} + {\omega_{D}n\;\tau} +} \\ {{\omega_{D}t_{1}} + \varphi_{0} + \zeta_{1} - {\omega_{M}n\;\tau}} \end{pmatrix}} -} \\ {j\;{\sin\begin{pmatrix} {{\omega_{R}n\;\tau} + {\omega_{R}t_{1}} + {\omega_{D}n\;\tau} +} \\ {{\omega_{D}t_{1}} + \varphi_{0} + \zeta_{1} - {\omega_{M}n\;\tau}} \end{pmatrix}}} \end{bmatrix}}}}} & (87) \end{matrix}$ Since ω_(M)≈ω_(R)+ω_(D) the terms of the sinusoids including n drop out.

When the update interval Δt₁≡1/f_(U) is given by Δt₁=m/f_(R) (where m is an integer) the value of t₁ is a multiple of 1/f_(R). Under this condition, for low stage velocities (i.e., ω_(D)<<ω_(R))ω_(R)t₁≈m2π and DFT calculation becomes:

$\begin{matrix} \begin{matrix} {{D\left( t_{1} \right)} \approx {A_{1}{\sum\limits_{n}{W_{n}\begin{bmatrix} {{\cos\left( {{\omega_{D}t_{1}} + \varphi_{0} + \zeta_{1}} \right)} -} \\ {j\;{\sin\left( {{\omega_{D}t_{1}} + \varphi_{0} + \zeta_{1}} \right)}} \end{bmatrix}}}}} \\ {\approx {A_{1}\left\lbrack {{\cos\left( {{\omega_{D}t_{1}} + \varphi_{0} + \zeta_{1}} \right)} - {j\;{\sin\left( {{\omega_{D}t_{1}} + \varphi_{0} + \zeta_{1}} \right)}}} \right\rbrack}} \end{matrix} & (89) \end{matrix}$ The DFT result is a complex value which can represent either a positive or a negative frequency. Equation (88) could be written as a sum, rather than a difference, with appropriate changes in sign in other equations. This choice of sign is not important, provided that the choices result in a reduction in cyclic error.

Equation (88) can be combined with equations (1) to (6). This results in: D(t ₁)=A ₁(cos(φ(t ₁)+ζ₁)−j sin(φ(t ₁)+ζ₁))+D _(ψ)(t ₁)  (89) where D _(ψ)(t ₁)=D _(ψ−1)(t ₁)+D _(ψ0) +D _(ψ2)(t ₁)+D _(ψ3)(t ₁);  (90) D _(ψ−1)(t ₁)=ε⁻¹(cos(−φ(t ₁)+ζ⁻¹)−j sin(−φ(t ₁)+ζ⁻¹)),  (91) D _(ψ0)=ε₀(cos(ζ₀)−j sin(ζ₀)),  (92) D _(ψ2)(t ₁)=ε₂(cos(2φ(t ₁)+₂)−j sin(2φ(t ₁)+ζ₂)),  (93) D _(ψ3)(t ₁)=ε₃(cos(3φ(t ₁)+ζ₃)−j sin(3φ(t ₁)+Λ₃));  (94) where φ(t₁) is the value of the Doppler phase shift φ(t₁)=ω_(D)t₁+φ₀ that is updated with t₁. For convenience, in the equations (98)-(116) below, t₁ is written simply as t and φ(t₁) is written simply as φ. Also, complex measurement signal D(t) 123 and complex error compensation signal D_(ψ)(t) 145 are assumed to be updated at the rate f_(U) such that D(t)≡D(t₁) and D_(ψ)(t)≡D_(ψ)(t₁).

FIG. 4 d shows a schematic diagram of CEC error estimator 144. In the following discussions the product of two complex signals frequently occurs. The following identities are used: (cos(A)+j sin(A))·(cos(B)+j sin(B))=cos(A+B)+j sin(A+B),  (95) (cos(A)+j sin(A))·(cos(B)−j sin(B))=cos(A−B)+j sin(A−B),  (96) (cos(A)+j sin(A))*=cos(A)−j sin(A)=cos(−A)+j sin(−A)  (97)

The next step is the processing of signals for information about the cyclic error terms. In a first “lowpass filtering approach,” the signal D(t) is sent to LPF (Lowpass Filter) 160, for example an IIR (Infinite Impulse Response) Butterworth digital filter, an FIR (Finite Impulse Response), or CIC (Cascaded Integrator Comb) digital filter as described by Hogenauer (An Economical Class of Digital Filters for Decimation and Interpolation; E. B. Hogenauer; IEEE Transactions on Acoustics, Speech, and Signal Processing; Vol ASSP-29, No 2, April 1981, p 155-162, incorporated herein by reference). The CIC filter has the advantages in this implementation of simple design (using only integer addition) and decimation by large ratios. The implementation of an LPF for a complex signal uses two identical real LPF functions, one is used for the real component, and one is used for the imaginary component. The use of digital functions ensures precise matching of amplitude and phase response of the two filters.

For an LPF T_(n)(x) of order n, the complex output C₀ is approximately:

$\begin{matrix} \begin{matrix} {C_{0} = {T_{n}\left\lbrack {D(t)} \right\rbrack}} \\ {= {{ɛ_{0}\left( {{\cos\;\left( \zeta_{0} \right)} - {j\;{\sin\left( \zeta_{0} \right)}}} \right)} +}} \\ {\left\lbrack {{A_{1}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} + {ɛ_{- 1}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} + {ɛ_{2}{O\left( \frac{\omega_{c}}{2\omega_{D}} \right)}^{n}} + {ɛ_{3}{O\left( \frac{\omega_{c}}{3\omega_{D}} \right)}^{n}}} \right\rbrack} \end{matrix} & (98) \end{matrix}$ where O(x) denotes a term of the order of x, ω_(c) is the −3 dB angular cutoff frequency, and ω_(D)=dφ/dt.

The term on the right hand sides of Equation (98) with factor A₁ is the source of the largest error and accordingly determines the specifications of n and the minimum ratio for ω_(D)/ω_(c) that can be used when the outputs of LPF 160 are stored in register 162. For a fourth order LPF, i.e., n=4, and a minimum ratio for ω_(D)/ω_(c)=7, the error terms on the right hand side of Equation (98) will generate errors that correspond to ≲0.010 nm (3σ). When the stage is moving at a speed such that the corresponding Doppler shift frequency ω_(D) /2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ω_(D)/ω_(c) is satisfied, the output C₀ of the LPF 160 is stored in register 162 as C_(0R) under the control of signal 161. This stage speed requirement reduces the possibility that sidebands of the primary Doppler signal caused by actual variations in the stage position or motion will be interpreted as cyclic errors.

An advantage of this approach is that ω_(D) can vary by factors such as 2 or more during the period when output values of C₀ are stored in register 162.

In a second approach to processing signals for information about the cyclic error terms, the CEC error estimator 144 stores values in the registers based on analysis of collective properties of a distribution of values of D(t). An advantage of this approach is that the stage can be nearly stationary, or moving at a speed such that the corresponding Doppler shift frequency ω_(D) /2π is less than 10 times greater than the bandwidth of the stage servo control system. In this case, the measured motion typically has negligible sidebands that could be interpreted as cyclic errors. This “distribution analysis approach” is described in more detail below.

The values for C_(0R) and D(t) are transmitted to processor 164 for the generation of complex signal Σ₁(t) where

$\begin{matrix} \begin{matrix} {{\Sigma_{1}(t)} = {{D(t)} - C_{0\; R}}} \\ {= {{A_{1}\left( {{\cos\left( {\varphi + \zeta_{1}} \right)} - {j\;\sin\;\left( {\varphi + \zeta_{- 1}} \right)}} \right)} +}} \\ {{ɛ_{- 1}\left( {{\cos\left( {{- \varphi} + \zeta_{- 1}} \right)} - {{jsin}\;\left( {{- \varphi} + \zeta_{- 1}} \right)}} \right)} +} \\ {{ɛ_{2}\left( {{\cos\left( {{2\varphi} + \zeta_{2}} \right)} - {{jsin}\;\left( {{2\varphi} + \zeta_{2}} \right)}} \right)} +} \\ {{ɛ_{3}\left( {{\cos\left( {{3\varphi} + \zeta_{3}} \right)} - {{jsin}\;\left( {{3\varphi} + \zeta_{3}} \right)}} \right)}.} \end{matrix} & (99) \end{matrix}$

Signal Σ₁ is sent to processor 168, which calculates Σ⁻¹ as the complex conjugate of Σ₁.

$\begin{matrix} \begin{matrix} {{\Sigma_{- 1}(t)} = {\Sigma_{1}(t)}^{*}} \\ {= {{A_{1}\left( {{\cos\left( {{- \varphi} - \zeta_{1}} \right)} - {j\;\sin\;\left( {{- \varphi} - \zeta_{1}} \right)}} \right)} +}} \\ {{ɛ_{- 1}\left( {{\cos\left( {\varphi - \zeta_{- 1}} \right)} - {{jsin}\;\left( {\varphi - \zeta_{- 1}} \right)}} \right)} +} \\ {{ɛ_{2}\left( {{\cos\left( {{{- 2}\varphi} - \zeta_{2}} \right)} - {{jsin}\;\left( {{{- 2}\varphi} - \zeta_{2}} \right)}} \right)} +} \\ {{ɛ_{3}\left( {{\cos\left( {{{- 3}\varphi} - \zeta_{3}} \right)} - {{jsin}\;\left( {{{- 3}\varphi} - \zeta_{3}} \right)}} \right)}.} \end{matrix} & (100) \end{matrix}$

Signal Σ₁ is sent to processor 180, which calculates Σ₂.

$\begin{matrix} \begin{matrix} {{\Sigma_{2}(t)} = \left( \Sigma_{1} \right)^{2}} \\ {= {{A_{1}^{2}\left( {{\cos\left( {{2\varphi} + {2\zeta_{1}}} \right)} - {j\;\sin\;\left( {{2\varphi} + {2\zeta_{1}}} \right)}} \right)} +}} \\ {{2A_{1}{ɛ_{- 1}\left( {{\cos\left( {\zeta_{1} + \zeta_{- 1}} \right)} - {{jsin}\;\left( {\zeta_{1} + \zeta_{- 1}} \right)}} \right)}} +} \\ {{2A_{1}{ɛ_{2}\left( {{\cos\left( {{3\varphi} + \zeta_{1} + \zeta_{2}} \right)} - {{jsin}\;\left( {{3\varphi} + \zeta_{1} + \zeta_{2}} \right)}} \right)}} +} \\ {{2A_{1}{ɛ_{3}\left( {{\cos\left( {{4\varphi} + \zeta_{1} + \zeta_{3}} \right)} - {{jsin}\;\left( {{4\varphi} + \zeta_{1} + \zeta_{3}} \right)}} \right)}} +} \\ {{O\left( {ɛ_{i},ɛ_{j}} \right)}.} \end{matrix} & (101) \end{matrix}$

Signal Σ₂ is divided by two and sent to LPF 190, as described earlier for LP 160.

For an LPF T_(n)(x) of order n, the complex output C₄ is approximately:

$\begin{matrix} \begin{matrix} {C_{4} = {T_{n}\left\lbrack {\frac{1}{2}\Sigma_{2}} \right\rbrack}} \\ {= {{A_{1}{ɛ_{- 1}\left( {{\cos\left( {\zeta_{1} + \zeta_{- 1}} \right)} - {{jsin}\left( {\zeta_{1} + \zeta_{- 1}} \right)}} \right)}} +}} \\ {\left\lbrack {{\frac{1}{2}A_{1}^{2}{O\left( \frac{\omega_{c}}{2\omega_{D}} \right)}^{n}} + {A_{1}ɛ_{2}{O\left( \frac{\omega_{c}}{3\omega_{D}} \right)}^{n}} + {A_{1}ɛ_{3}{O\left( \frac{\omega_{c}}{4\omega_{D}} \right)}^{n}}} \right\rbrack} \end{matrix} & (102) \end{matrix}$ where O(x) denotes a term of the order of x, ω_(c) is the −3 dB angular cutoff frequency, and ω_(D)=dφ/dt.

The term on the right hand sides of Equation (102) with factors A₁ ² is the source of the largest error and accordingly determines the specifications of n and the minimum ratio for ω_(D)/ω_(c) that can be used when the outputs of processor 190 are stored in register 192. For a fourth order LPF, i.e., n=4, and a minimum ratio for ω_(D)/ω_(c)=3.5, the error terms on the right hand side of Equation (102) will generate errors that correspond to ≲0.010 nm (3σ). When the stage is moving at a speed such that the corresponding Doppler shift frequency ω_(D) /2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ω_(D)/ω_(c) is satisfied, the output C₄ of LPF 190 is stored in register 192 as C_(4R) under the control of signal 161.

Signals Σ₁, Σ₂, and C_(4R) are sent to processor 200 which calculates Σ₃. Signal Σ_(3A) is calculated by combining signals C_(4R) and Σ₁ using multiplier 194:

$\begin{matrix} \begin{matrix} {\Sigma_{3A} = {C_{4R}\Sigma_{1}}} \\ {= {{A_{1}^{2}{ɛ_{- 1}\left( {{\cos\left( {{2\zeta_{1}} + \zeta_{- 1} + \varphi} \right)} - {{jsin}\left( {{2\zeta_{1}} + \zeta_{- 1} + \varphi} \right)}} \right)}} +}} \\ {{A_{1}{ɛ_{- 1}^{2}\left( {{\cos\left( {\zeta_{1} + {2\zeta_{- 1}} - \varphi} \right)} - {{jsin}\left( {\zeta_{1} + {2\zeta_{- 1}} - \varphi} \right)}} \right)}} +} \\ {A_{1}ɛ_{- 1}{ɛ_{2}\left( {{\cos\left( {\zeta_{1} + \zeta_{- 1} + {2\varphi} + \zeta_{2}} \right)} -} \right.}} \\ {\left. {{jsin}\left( {\zeta_{1} + \zeta_{- 1} + {2\varphi} + \zeta_{2}} \right)} \right) +} \\ {A_{1}ɛ_{- 1}{ɛ_{3}\left( {{\cos\left( {\zeta_{1} + \zeta_{- 1} + {3\varphi} + \zeta_{3}} \right)} -} \right.}} \\ {\left. {{jsin}\left( {\zeta_{1} + \zeta_{- 1} + {3\varphi} + \zeta_{3}} \right)} \right) + {A_{1}ɛ_{- 1}{{O\left( {ɛ_{i},ɛ_{j}} \right)}.}}} \end{matrix} & (103) \end{matrix}$

Signal Σ_(3B) is calculated by combining signals Σ₁ and Σ₂ using multiplier 202:

$\begin{matrix} \begin{matrix} {\Sigma_{3B} = {\Sigma_{1}\Sigma_{2}}} \\ {= {{A_{1}^{3}\left( {{\cos\left( {{3\varphi} + {3\zeta_{1}}} \right)} - {{jsin}\left( {{3\varphi} + {3\zeta_{1}}} \right)}} \right)} +}} \\ {{3A_{1}^{2}{ɛ_{- 1}\left( {{\cos\left( {\varphi + {2\zeta_{1}} + \zeta_{- 1}} \right)} - {{jsin}\left( {\varphi + {2\zeta_{1}} + \zeta_{- 1}} \right)}} \right)}} +} \\ {{3A_{1}^{2}{ɛ_{2}\left( {{\cos\left( {{4\varphi} + {2\zeta_{1}} + \zeta_{2}} \right)} - {{jsin}\left( {{4\varphi} + {2\zeta_{1}} + \zeta_{2}} \right)}} \right)}} +} \\ {{3A_{1}^{2}{ɛ_{3}\left( {{\cos\left( {{5\varphi} + {2\zeta_{1}} + \zeta_{3}} \right)} - {{jsin}\left( {{5\varphi} + {2\zeta_{1}} + \zeta_{3}} \right)}} \right)}} +} \\ {{O\left( {ɛ_{i},ɛ_{j}} \right)}.} \end{matrix} & (104) \end{matrix}$

Signal Σ₃ is calculated by combining signals Σ_(3A) and Σ_(3B) using constant multiplier 196 and subtractor 198:

$\begin{matrix} \begin{matrix} {\Sigma_{3} = {\Sigma_{3B} - {3\Sigma_{3A}}}} \\ {= {{A_{1}^{3}\left( {{\cos\left( {{3\varphi} + {3\zeta_{1}}} \right)} - {{jsin}\left( {{3\varphi} + {3\zeta_{1}}} \right)}} \right)} +}} \\ {{3A_{1}^{2}{ɛ_{2}\left( {{\cos\left( {{4\varphi} + {2\zeta_{1}} + \zeta_{2}} \right)} - {{jsin}\left( {{4\varphi} + {2\zeta_{1}} + \zeta_{2}} \right)}} \right)}} +} \\ {{3A_{1}^{2}{ɛ_{3}\left( {{\cos\left( {{5\varphi} + {2\zeta_{1}} + \zeta_{3}} \right)} - {{jsin}\left( {{5\varphi} + {2\zeta_{1}} + \zeta_{3}} \right)}} \right)}} +} \\ {3A_{1}{{O\left( {ɛ_{i},ɛ_{j}} \right)}.}} \end{matrix} & (105) \end{matrix}$

Signal Σ₅ is calculated by combining signals Σ⁻¹ and Σ₂ using multiplier 182:

$\begin{matrix} \begin{matrix} {\Sigma_{5} = {\Sigma_{- 1}\Sigma_{2}}} \\ {= {{A_{1}^{2}{ɛ_{2}\left( {{\cos\left( {{2\zeta_{1}} - \xi_{2}} \right)} - {{jsin}\left( {{2\zeta_{1}} - \zeta_{2}} \right)}} \right)}} +}} \\ {{A_{1}^{2}\begin{Bmatrix} {{+ {ɛ_{- 1}\begin{pmatrix} {{2\left( {{\cos\left( {{- \varphi} + \zeta_{- 1}} \right)} - {{jsin}\left( {{- \varphi} - \zeta_{1}} \right)}} \right)} +} \\ \left( {{\cos\left( {{3\varphi} + {2\zeta_{1}} - \zeta_{- 1}} \right)} -} \right. \\ \left. {{jsin}\left( {{3\varphi} + {2\zeta_{1}} - \zeta_{- 1}} \right)} \right) \end{pmatrix}}} +} \\ {{A_{1}\left( {{\cos\left( {\varphi + \zeta_{1}} \right)} - {{jsin}\left( {\varphi + \zeta_{1}} \right)}} \right)} +} \\ {{2{ɛ_{2}\left( {{\cos\left( {{2\varphi} + \zeta_{2}} \right)} - {{jsin}\left( {{2\varphi} + \zeta_{2}} \right)}} \right)}} +} \\ {ɛ_{3}\begin{pmatrix} {{2\left( {{\cos\left( {{3\varphi} + \zeta_{3}} \right)} - {{jsin}\left( {{3\varphi} + \zeta_{3}} \right)}} \right)} +} \\ \left( {{\cos\left( {{- \varphi} + {2\zeta_{1}} - \zeta_{3}} \right)} - {{jsin}\left( {{- \varphi} + {2\zeta_{1}} - \zeta_{3}} \right)}} \right) \end{pmatrix}} \end{Bmatrix}} +} \\ {{A_{1}{O\left( ɛ_{i} \right)}} + {\ldots\mspace{11mu}.}} \end{matrix} & (106) \end{matrix}$

Signal Σ₅ is sent to LPF 184, as described earlier for LPF 160.

For an LPF T_(n)(x) of order n, the complex output C₅ is approximately:

$\begin{matrix} \begin{matrix} {C_{5} = {T_{n}\left\lbrack \Sigma_{5} \right\rbrack}} \\ {= {{A_{1}^{2}{ɛ_{2}\left( {{\cos\left( {{2\zeta_{1}} - \zeta_{2}} \right)} - {{jsin}\left( {{2\zeta_{1}} - \zeta_{2}} \right)}} \right)}} +}} \\ {A_{1}^{2}\begin{bmatrix} {{2ɛ_{- 1}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} + {ɛ_{- 1}{O\left( \frac{\omega_{c}}{3\omega_{D}} \right)}^{n}} + {A_{1}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} +} \\ {{2ɛ_{2}{O\left( \frac{\omega_{c}}{2\omega_{D}} \right)}^{n}} + {2ɛ_{3}{O\left( \frac{\omega_{c}}{3\omega_{D}} \right)}^{n}} + {ɛ_{3}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}}} \end{bmatrix}} \end{matrix} & (107) \end{matrix}$ where O(x) denotes a term of the order of x, ω_(c) is the −3 dB angular cutoff frequency, and ω_(D)=dφ/dt.

The term on the right hand sides of Equation (107) with factor A₁ ³ is the source of the largest errors and accordingly determines the specifications of n and the minimum ratio for ω_(D)/ω_(c) that can be used when the outputs of LPF 184 is stored in register 186. For a fourth order LPF, i.e., n=4, and a minimum ratio for ω_(D)/ω_(c)=7, the error terms on the right hand side of Equation (107) will generate errors that correspond to ≲0.010 nm (3σ). When the stage is moving at a speed such that the corresponding Doppler shift frequency ω_(D) /2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ω_(D)/ω_(c) is satisfied, the output C₅ of LPF 184 is stored in register 186 as C_(5R) under the control of signal 161.

Signal Σ₆ is calculated by combining signals Σ⁻¹ and Σ₃ using multiplier 172:

$\begin{matrix} \begin{matrix} {\Sigma_{6} = {\Sigma_{- 1}\Sigma_{3}}} \\ {= {{A_{1}^{3}{ɛ_{3}\left( {{\cos\left( {{3\zeta_{1}} - \zeta_{3}} \right)} - {{jsin}\left( {{3\zeta_{1}} - \zeta_{3}} \right)}} \right)}} +}} \\ {{A_{1}^{3}{ɛ_{- 1}\left( {{\cos\left( {{4\varphi} - \zeta_{- 1} + {3\zeta_{1}}} \right)} - {{jsin}\left( {{4\varphi} - \zeta_{- 1} + {3\zeta_{1}}} \right)}} \right)}} +} \\ {{A_{1}^{4}\left( {{\cos\left( {{2\varphi} + {2\zeta_{1}}} \right)} - {{jsin}\left( {{2\varphi} + {2\zeta_{1}}} \right)}} \right)} +} \\ {{A_{1}^{3}{ɛ_{2}\begin{pmatrix} {{3\left( {{\cos\left( {{3\varphi} + \zeta_{1} + \zeta_{2}} \right)} - {{jsin}\left( {{3\varphi} + \zeta_{1} + \zeta_{2}} \right)}} \right)} +} \\ \left( {{\cos\left( {\varphi + {3\zeta_{1}} - \zeta_{2}} \right)} - {{jsin}\left( {\varphi + {3\zeta_{1}} - \zeta_{2}} \right)}} \right) \end{pmatrix}}} +} \\ {{3A_{1}^{3}{ɛ_{3}\left( {{\cos\left( {{4\varphi} + \zeta_{1} + \zeta_{3}} \right)} - {j\;{\sin\left( {{4\varphi} + \zeta_{1} + \zeta} \right)}}} \right)}} +} \\ {{3A_{1}^{2}{O\left( {ɛ_{i}ɛ_{j}} \right)}} + {\ldots\mspace{11mu}.}} \end{matrix} & (108) \end{matrix}$

Signal Σ₆ is sent to LPF 174, as described earlier for LPF 160.

For an LPF T_(n)(x) of order n, the complex output C₆ is approximately:

$\begin{matrix} \begin{matrix} {C_{6} = {T_{n}\left\lbrack \Sigma_{6} \right\rbrack}} \\ {= {{A_{1}^{3}{ɛ_{3}\left( {{\cos\left( {{3\zeta_{1}} - \zeta_{3}} \right)} - {{jsin}\left( {{3\zeta_{1}} - \zeta_{3}} \right)}} \right)}} +}} \\ {A_{1}^{3}\begin{bmatrix} {{ɛ_{- 1}{O\left( \frac{\omega_{c}}{4\omega_{D}} \right)}^{n}} + {A_{1}{O\left( \frac{\omega_{c}}{2\omega_{D}} \right)}^{n}} +} \\ {{3ɛ_{2}{O\left( \frac{\omega_{c}}{3\omega_{D}} \right)}^{n}} + {ɛ_{2}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} + {3ɛ_{3}{O\left( \frac{\omega_{c}}{4\omega_{D}} \right)}^{n}}} \end{bmatrix}} \end{matrix} & (109) \end{matrix}$ where O(x) denotes a term of the order of x, ω_(c) is the −3 dB angular cutoff frequency, and ω_(D)=dφ/dt.

The term on the right hand sides of Equation (107) with factor A₁ ⁴ is the source of the largest errors and accordingly determines the specifications of n and the minimum ratio for ω_(D)/ω_(c) that can be used when the outputs of LPF 174 is stored in register 176. For a fourth order LPF, i.e., n=4, and a minimum ratio for ω_(D)/ω_(c)=3.5, the error terms on the right hand side of Equation (109) will generate errors that correspond to ≲0.010 nm (3σ). When the stage is moving at a speed such that the corresponding Doppler shift frequency ω_(D)/2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ω_(D)/ω_(c) is satisfied, the output C₆ of LPF 174 is stored in register 176 as C_(6R) under the control of signal 161.

The magnitude squared of signal D(t) is calculated by multiplier 152: C ₁(t)=D(t)·D(t)*.  (110)

This may be simplified and implemented as: C ₁(t)=Re(D(t))² +Im(D(t))².  (111)

Alternate methods to calculate a value equivalent to the result of equation (111) are possible, for example the magnitude output 125 from CORDIC converter 124 could be squared, or signal Σ₁ could be used instead of D(t).

When equation (111) is evaluated using the terms of D(t) from equations (89) to (94), the following result is obtained:

$\begin{matrix} {{C_{1}(t)} = {A_{1}^{2} + \left\lbrack {ɛ_{- 1}^{2} + ɛ_{0}^{2} + ɛ_{2}^{2} + ɛ_{3}^{2}} \right\rbrack + {2A_{1}ɛ_{- 1}{\cos\left( {{2\varphi} + \zeta_{1} - \zeta_{- 1}} \right)}} + {2A_{1}ɛ_{0}{\cos\left( {\varphi + \zeta_{1} - \zeta_{0}} \right)}} + {2A_{1}ɛ_{2}{\cos\left( {{- \varphi} + \zeta_{1} - \zeta_{2}} \right)}} + {2A_{1}ɛ_{3}{\cos\left( {{{- 2}\varphi} + \zeta_{1} - \zeta_{3}} \right)}} + {{O\left( {ɛ_{i}ɛ_{j}} \right)}.}}} & (112) \end{matrix}$

The signal C₁(t) is sent to LPF (Lowpass Filter) 154 as described earlier for LPF 160.

For an LPF T_(n)(x) of order n, the output C₁ is approximately:

$\begin{matrix} \begin{matrix} {C_{1} = {T_{n}\left\lbrack {C_{1}(t)} \right\rbrack}} \\ {= {A_{1}^{2} + \left\lbrack {ɛ_{- 1}^{2} + ɛ_{0}^{2} + ɛ_{2}^{2} + ɛ_{3}^{2}} \right\rbrack + {2A_{1}} + {\begin{bmatrix} {{ɛ_{- 1}{O\left( \frac{\omega_{c}}{2\omega_{D}} \right)}^{n}} + {ɛ_{0}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} +} \\ {{ɛ_{2}{O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}} + {ɛ_{3}{O\left( \frac{\omega_{c}}{2\omega_{D}} \right)}^{n}}} \end{bmatrix}.}}} \end{matrix} & (113) \end{matrix}$

The accuracy required for the determination of C₁ is approximately 0.5% in order to limit errors generated in the computation of cyclic error signals S_(ψj) to ≲0.010 nm (3σ). Therefore the error terms ε⁻¹ ², ε₀ ², ε₂ ², and ε₂ ² on the right hand side of Equation (113) are negligible. The terms on the right hand side of Equation (113) of the form

${O\left( \frac{\omega_{c}}{\omega_{D}} \right)}^{n}$ are the sources of the largest Doppler shift frequency dependent errors and accordingly determine the specifications of n and the minimum ratio for ω_(D)/ω_(c) that can be used when the output of LPF 154 is held in register 156, providing signal C_(1R). For a second order LPF, i.e., n=2, and a minimum ratio for ω_(D)/ω_(c)=3.5, the Doppler shift frequency dependent error terms on the right hand side of Equation (113) will generate errors that correspond to ≲0.010 nm (3σ). When the stage is moving at a speed such that the corresponding Doppler shift frequency ω_(D)/2π is 10 to 100 times greater than the bandwidth of the stage servo control system and the requirement with respect to ω_(D)/ω_(c) is satisfied, the output C₁ of LPF 154 is stored in register 156 as C_(1R) under the control of signal 161.

The low pass filtering approach to determining values from which error basis functions and their coefficients are derived is appropriate when the stage is moving at a speed such that the corresponding Doppler shift frequency satisfies constraints due to the low pass filter −3 dB cutoff ω_(c) and the stage servo control system bandwidth. When the stage is being scanned slowly, or is nearly stationary (e.g., due to a change in scan direction), a distribution analysis approach can be used to calculate and store the values used to generate the compensating signal D_(ψ)(t).

The distribution analysis approach includes performing error compensation calculations based on collective properties of a distribution of values derived at least in part from samples of the signal S(t). The values may represent, for example, samples of a multi-dimensional signal. As a function of time, the multi-dimensional signal defines a curve, and the values represent points on the curve. The distribution analysis approach is described in more detail, for example, in U.S. application Ser. No. 11/462,185, incorporated herein by reference.

Using the coefficients derived using either the low pass filtering approach or the distribution analysis approach, the processor 204 calculates compensating signal D_(ψ)(t) as shown in FIG. 4 e and Equation (114).

$\begin{matrix} {{D\;{\psi(t)}} = {C_{0R} + {\frac{C_{4R}}{C_{1R}}\Sigma_{- 1}} + {\left( {\frac{C_{5R}}{C_{1R}^{2}}*}\; \right)\Sigma_{2}} + {\left( {\frac{C_{6R}}{C_{1R}^{3}}*}\; \right)\Sigma_{3}}}} & (118) \end{matrix}$

This can be expanded using the most significant terms from the previously expressed equations (98), (113), (102), (107), and (109) for the C_(N) terms, and equations (100), (101), and (105) for the Σ_(N) terms to:

$\begin{matrix} {{D\;{\psi(t)}} = {{ɛ_{0}\left( {{\cos\left( \zeta_{0} \right)} - {{jsin}\left( \zeta_{0} \right)}} \right)} + {\frac{A_{1}{ɛ_{- 1}\left( {{\cos\left( {\zeta_{1} + \zeta_{- 1}} \right)} - {{jsin}\left( {\zeta_{1} + \zeta_{- 1}} \right)}} \right)}}{A_{1}^{2}}{A_{1}\left( {{\cos\left( {{- \varphi} - \zeta_{1}} \right)} - {{jsin}\left( {{- \varphi} - \zeta_{1}} \right)}} \right)}} + {\left( {\frac{A_{1}^{2}{ɛ_{2}\left( {{\cos\left( {{2\zeta_{1}} - \zeta_{2}} \right)} - {{jsin}\left( {{2\zeta_{1}} - \zeta_{2}} \right)}} \right)}}{A_{1}^{4}}*}\; \right){A_{1}^{2}\left( {{\cos\left( {{2\varphi} + {2\zeta_{1}}} \right)} - {{jsin}\left( {{2\varphi} + {2\zeta_{1}}} \right)}} \right)}} + {\left( {\frac{A_{1}^{3}{ɛ_{3}\left( {{\cos\left( {{3\zeta_{1}} - \zeta_{3}} \right)} - {{jsin}\left( {{3\zeta_{1}} - \zeta_{3}} \right)}} \right)}}{A_{1}^{6}}*}\; \right){A_{1}^{3}\left( {{\cos\left( {{3\varphi} + {3\zeta_{1}}} \right)} - {{jsin}\left( {{3\varphi} + {3\zeta_{1}}} \right)}} \right)}}}} & (119) \end{matrix}$ and simplified to:

$\begin{matrix} {{D\;{\psi(t)}} = {{ɛ_{0}\left( {{\cos\left( \zeta_{0} \right)} - {{jsin}\left( \zeta_{0} \right)}} \right)} + {ɛ_{- 1}\left( {{\cos\left( {{- \varphi} + \zeta_{- 1}} \right)} - {{jsin}\left( {{- \varphi} + \zeta_{- 1}} \right)}} \right)} + {ɛ_{2}\left( {{\cos\left( {{2\varphi} + \zeta_{2}} \right)} - {{jsin}\left( {{2\varphi} + \zeta_{2}} \right)}} \right)} + {{ɛ_{3}\left( {{\cos\left( {{3\varphi} + \zeta_{3}} \right)} - {{jsin}\left( {{3\varphi} + \zeta_{3}} \right)}} \right)}.}}} & (120) \end{matrix}$

Note that the calculated result in equation (116) is identical to D_(ψ)(t) as described in equations (90) to (94).

Control signal 211 enables the output of D_(ψ)(t) when conditions indicate that a valid compensation signal is present. Control signal 211 is generated by control circuitry (not shown) that may consider the speed of motion, the signal level, the cyclic error level, the duration of motion, and possible other conditions to determine the validity or usefulness of D_(ψ)(t). It may also be desirable to enable each of the four cyclic error terms separately, for example if the ε₃ term was found to be insignificant for a specific implementation, the calculation could be disabled to reduce noise contributed by the calculation. When a D_(ψ)(t) term or output is disabled, zero is substituted.

The calculated D_(ψ)(t) is subtracted from signal D(t) by subtractor 146, resulting in a measurement signal with substantially reduced cyclic errors.

Using either the DMS approach or the CMS approach, other techniques can be incorporated to compensate for some of the sources of cyclic errors. For example, the dominant error term is typically the unshifted cyclic error component, S_(ψ0), or equivalently D_(ψ0), which stays at constant phase and frequency regardless of stage motion. This term arises from the presence of both optical frequencies in either the reference arm or the measurement arm of the displacement measuring interferometer or both. This occurs, for example, if the optical frequencies of the light source are not perfectly separated into orthogonal linear polarization states.

Though the cyclic error compensation techniques are described using the double pass plane mirror interferometer by way of example, they can be applied to any two-frequency, displacement measuring interferometer in which the cyclic error term which does not experience Doppler shift is dominant.

Compensation for this unshifted term can be achieved by momentarily blocking the beam in the reference path of the interferometer and then in the measurement path of the interferometer at two different times, during, for example, interferometer installation; measuring and recording the complex amplitude of the measurement signal in the beam-blocked condition; adding the two different complex amplitudes so obtained to obtain a compensation coefficient representing the unshifted term that is present when neither beam is blocked; then using the complex coefficient to compensate for the unshifted term during operation of the electronic compensation techniques described herein with the automatic coefficient updating disabled.

The technique of beam blocking can either be manual or automated using, for example, motorized shutters in the reference and measurement paths of the interferometer which are controlled by the same computer that reads the complex amplitudes in the beam-blocked condition; calculates the sum; and stores the coefficient for use in the electronic compensation techniques described herein.

Finally, it is noted that, if desired, the accuracy of the cyclic error correction can further be improved to higher order in the amplitude of the cyclic error coefficients by iterating the compensation of the main signal S(t). In other words, for each subsequent iteration the compensated signal for the main signal is used to generate corresponding cyclic error basis functions and determine a higher order correction to each of the cyclic error coefficients.

Referring now to FIG. 5, a block diagram is shown for an M^(th) order digital filter used in the cyclic error compensation described above. The figure is in the “Direct Form I” representation standard to the digital signal processing community. The input discrete time series is x(n) and the output discrete time series is y(n). The z⁻¹ operation indicates a one-sample delay. A time-domain representation of the filter takes the form: y(n)=b ₀ x(n)+b ₁ x(n−1)+b ₂ x(n−2)+ . . . +b _(M) x(n−M)−a ₁ y(n−1)−a ₂ y(n−2)− . . . −a _(M) y(n−M)  (121). The coefficients a_(i) and b_(i) are selected to produce the desired properties for the filter. For the case of the Butterworth filter, the a_(i) and b_(i) coefficients are selected to produce the frequency filtering given by Equations 28, 29, 43, 44, 55, and 56. Furthermore, other embodiments of the cyclic error compensation may implement different low-pass filtering schemes to yield the coefficients of low-frequency terms. The Butterworth filter, and other low-pass digital filters, are well known in the art. See, for example: Oppenhiem, A. V., Schafer, R. W., and, J. R. Buck, “Discrete-Time Signal Processing”, Upper Saddle River, N.J.: Prentice Hall, 1999; and Proakis, J. G., and, D. G. Manolakis, “Digital Signal Processing: Principles, Algorithms, and Applications”, New York, N.Y.: MacMillan, 1992.

Depending on the embodiment, the compensation technique described above can be implemented in hardware or software, or a combination of both. The technique can be implemented in computer programs using standard programming techniques following the method and figures described herein. Program code is applied to input data to perform the functions described herein and generate output information. The output information is applied to one or more output devices such as the servo control system.

Each program may be implemented in a high level procedural or object oriented programming language to communicate with a computer system, or the programs can be implemented in assembly or machine language, if desired. In any case, the language can be a compiled or interpreted language. Moreover, the program can run on dedicated integrated circuits preprogrammed for that purpose.

Each such computer program may be stored on a storage medium or device (e.g., ROM or magnetic diskette) readable by a general or special purpose programmable computer, for configuring and operating the computer when the storage media or device is read by the computer to perform the procedures described herein. The computer program can also reside in cache or main memory during program execution. The compensation technique can also be implemented as a computer-readable storage medium, configured with a computer program, where the storage medium so configured causes a computer to operate in a specific and predefined manner to perform the functions described herein.

Now referring to FIG. 6, an interferometry system including a high stability plane mirror interferometer (HSPMI) 411 is shown for optical generating the main interference signal. The HSPMI 411 includes a polarization beam-splitter 330, a retroreflector 332, quarter wave phase retardation plates 334 and 336, and a plane mirror reference object 342. Input beam 422 is a two-component beam. The two components have different frequencies and are orthogonally plane polarized. The different frequencies can be produced in source 415, for example, by laser Zeeman splitting, by acousto-optical modulation, or internal to the laser using birefringent elements or the like. HSPMI 411 splits input beam 422 into two components. One component, shown as first and second pass measurement beams 322 and 324, reflects from measurement object 490 twice before exiting HSPMI 411. The other component, shown by first and second pass reference beams 328 and 327, reflect from reference mirror 342 twice before exiting HSPMI 411. The exiting beam components overlap and form output beam 423.

An electrical interference signal 352 is generated by the detection of output beam 423 in detector 420. Detector 420 includes a polarizer to mix the reference and measurement beam components of output beam 423 with respect to polarization. Electrical interference signal 352 contains a heterodyne interference signal corresponding to main interference signal S(t).

In further embodiments, the interferometry system may be different than that shown in FIG. 6. In particular, the cyclic error compensation technique is applicable to other types of interferometers as well. For example, the main interference signal S(t) may be produced by an interferometry system that may include any of single and/or multiple pass interferometers, passive interferometers, dynamic interferometers, and dispersion interferometers. Furthermore, the interferometry system may monitor one or more degrees of freedom, each of which may produce a corresponding main interference signal S(t), which may be compensated for cyclic errors as disclosed herein. Furthermore, the degree(s) of freedom monitored by the interferometry system may include any of changes in distance to a measurement object, changes in relative distance between two measurement objects, changes in the angular orientation of a measurement object, and changes in the direction of the input beam.

Examples of dynamic interferometers are described in U.S. patent application Ser. No. 10/226,591 filed Aug. 23, 2002 and entitled “DYNAMIC INTERFEROMETER CONTROLLING DIRECTION OF INPUT BEAM” by Henry A. Hill. Examples of passive zero shear interferometers are described in U.S. patent application Ser. No. 10/207,314, entitled “PASSIVE ZERO SHEAR INTERFEROMETERS,” filed Jul. 29, 2002, by Henry A. Hill. Examples of angular displacement interferometers are described in: U.S. patent application Ser. No. 10/226,591 entitled “DYNAMIC INTERFEROMETER CONTROLLING DIRECTION OF INPUT BEAM,” filed Aug. 23, 2002; U.S. Provisional Application 60/314,345 filed Aug. 22, 2001 and entitled “PASSIVE ZERO SHEAR INTERFEROMETERS USING ANGLE SENSITIVE BEAM-SPLITTERS,” both by Henry A. Hill, and U.S. patent application Ser. No. 10/272,034 entitled “INTERFEROMETERS FOR MEASURING CHANGES IN OPTICAL BEAM DIRECTION” and filed Oct. 15, 2002 by Henry A. Hill and Justin Kreuzer. Alternatively, or additionally, interferometry systems may include one or more differential angular displacement interferometers, examples of which are also described in U.S. patent application Ser. No. 10/272,034. Examples of interferometry systems for measuring more than one degree of freedom and for reducing beam shear are described in U.S. patent application Ser. No. 10/352,616 filed Jan. 28, 2003 and entitled “MULTIPLE-PASS INTERFEROMETRY” by Henry A. Hill and U.S. patent application Ser. No. 10/351,708 filed Jan. 27, 2003 and entitled “MULTI-AXIS INTERFEROMETER” by Henry A. Hill. Other forms of multiple pass interferometers are described in an article entitled “Differential interferometer arrangements for distance and angle measurements: Principles, advantages and applications” by C. Zanoni, VDI Berichte Nr. 749, 93-106 (1989). Examples of two-wavelength dispersion interferometers are described in U.S. Pat. No. 6,219,144 B1 entitled “APPARATUS AND METHOD FOR MEASURING THE REFRACTIVE INDEX AND OPTICAL PATH LENGTH EFFECTS OF AIR USING MULTIPLE-PASS INTERFEROMETRY” by Henry A. Hill, Peter de Groot, and Frank C. Demarest and U.S. Pat. No. 6,327,039 B1 by Peter de Groot, Henry A. Hill, and Frank C. Demarest.

Because of the cyclic error compensation, the interferometry systems described herein provide highly accurate measurements. Such systems can be especially useful in lithography applications used in fabricating large scale integrated circuits such as computer chips and the like. Lithography is the key technology driver for the semiconductor manufacturing industry. Overlay improvement is one of the five most difficult challenges down to and below 100 nm line widths (design rules), see, for example, the Semiconductor Industry Roadmap, p. 82 (1997).

Overlay depends directly on the performance, i.e., accuracy and precision, of the distance measuring interferometers used to position the wafer and reticle (or mask) stages. Since a lithography tool may produce $50-100M/year of product, the economic value from improved performance distance measuring interferometers is substantial. Each 1% increase in yield of the lithography tool results in approximately $1M/year economic benefit to the integrated circuit manufacturer and substantial competitive advantage to the lithography tool vendor.

The function of a lithography tool is to direct spatially patterned radiation onto a photoresist-coated wafer. The process involves determining which location of the wafer is to receive the radiation (alignment) and applying the radiation to the photoresist at that location (exposure).

To properly position the wafer, the wafer includes alignment marks on the wafer that can be measured by dedicated sensors. The measured positions of the alignment marks define the location of the wafer within the tool. This information, along with a specification of the desired patterning of the wafer surface, guides the alignment of the wafer relative to the spatially patterned radiation. Based on such information, a translatable stage supporting the photoresist-coated wafer moves the wafer such that the radiation will expose the correct location of the wafer.

During exposure, a radiation source illuminates a patterned reticle, which scatters the radiation to produce the spatially patterned radiation. The reticle is also referred to as a mask, and these terms are used interchangeably below. In the case of reduction lithography, a reduction lens collects the scattered radiation and forms a reduced image of the reticle pattern. Alternatively, in the case of proximity printing, the scattered radiation propagates a small distance (typically on the order of microns) before contacting the wafer to produce a 1:1 image of the reticle pattern. The radiation initiates photo-chemical processes in the resist that convert the radiation pattern into a latent image within the resist.

Interferometry systems are important components of the positioning mechanisms that control the position of the wafer and reticle, and register the reticle image on the wafer. If such interferometry systems include the features described above, the accuracy of distances measured by the systems increases as cyclic error contributions to the distance measurement are minimized.

In general, the lithography system, also referred to as an exposure system, typically includes an illumination system and a wafer positioning system. The illumination system includes a radiation source for providing radiation such as ultraviolet, visible, x-ray, electron, or ion radiation, and a reticle or mask for imparting the pattern to the radiation, thereby generating the spatially patterned radiation. In addition, for the case of reduction lithography, the illumination system can include a lens assembly for imaging the spatially patterned radiation onto the wafer. The imaged radiation exposes resist coated onto the wafer. The illumination system also includes a mask stage for supporting the mask and a positioning system for adjusting the position of the mask stage relative to the radiation directed through the mask. The wafer positioning system includes a wafer stage for supporting the wafer and a positioning system for adjusting the position of the wafer stage relative to the imaged radiation. Fabrication of integrated circuits can include multiple exposing steps. For a general reference on lithography, see, for example, J. R. Sheats and B. W. Smith, in Microlithography: Science and Technology (Marcel Dekker, Inc., New York, 1998), the contents of which is incorporated herein by reference.

Interferometry systems described above can be used to precisely measure the positions of each of the wafer stage and mask stage relative to other components of the exposure system, such as the lens assembly, radiation source, or support structure. In such cases, the interferometry system can be attached to a stationary structure and the measurement object attached to a movable element such as one of the mask and wafer stages. Alternatively, the situation can be reversed, with the interferometry system attached to a movable object and the measurement object attached to a stationary object.

More generally, such interferometry systems can be used to measure the position of any one component of the exposure system relative to any other component of the exposure system, in which the interferometry system is attached to, or supported by, one of the components and the measurement object is attached, or is supported by the other of the components.

An example of a lithography scanner 1100 using an interferometry system 1126 is shown in FIG. 7. The interferometry system is used to precisely measure the position of a wafer (not shown) within an exposure system. Here, stage 1122 is used to position and support the wafer relative to an exposure station. Scanner 1100 includes a frame 1102, which carries other support structures and various components carried on those structures. An exposure base 1104 has mounted on top of it a lens housing 1106 atop of which is mounted a reticle or mask stage 1116, which is used to support a reticle or mask. A positioning system for positioning the mask relative to the exposure station is indicated schematically by element 1117. Positioning system 1117 can include, e.g., piezoelectric transducer elements and corresponding control electronics. Although, it is not included in this described embodiment, one or more of the interferometry systems described above can also be used to precisely measure the position of the mask stage as well as other moveable elements whose position must be accurately monitored in processes for fabricating lithographic structures (see supra Sheats and Smith Microlithography: Science and Technology).

Suspended below exposure base 1104 is a support base 1113 that carries wafer stage 1122. Stage 1122 includes a plane mirror 1128 for reflecting a measurement beam 1154 directed to the stage by interferometry system 1126. A positioning system for positioning stage 1122 relative to interferometry system 1126 is indicated schematically by element 1119. Positioning system 1119 can include, e.g., piezoelectric transducer elements and corresponding control electronics. The measurement beam reflects back to the interferometry system, which is mounted on exposure base 1104. The interferometry system can be any of the embodiments described previously.

During operation, a radiation beam 1110, e.g., an ultraviolet (UV) beam from a UV laser (not shown), passes through a beam shaping optics assembly 1112 and travels downward after reflecting from mirror 1114. Thereafter, the radiation beam passes through a mask (not shown) carried by mask stage 1116. The mask (not shown) is imaged onto a wafer (not shown) on wafer stage 1122 via a lens assembly 1108 carried in a lens housing 1106. Base 1104 and the various components supported by it are isolated from environmental vibrations by a damping system depicted by spring 1120.

In other embodiments of the lithographic scanner, one or more of the interferometry systems described previously can be used to measure distance along multiple axes and angles associated for example with, but not limited to, the wafer and reticle (or mask) stages. Also, rather than a UV laser beam, other beams can be used to expose the wafer including, e.g., x-ray beams, electron beams, ion beams, and visible optical beams.

In some embodiments, the lithographic scanner can include what is known in the art as a column reference. In such embodiments, the interferometry system 1126 directs the reference beam (not shown) along an external reference path that contacts a reference mirror (not shown) mounted on some structure that directs the radiation beam, e.g., lens housing 1106. The reference mirror reflects the reference beam back to the interferometry system. The interference signal produce by interferometry system 1126 when combining measurement beam 1154 reflected from stage 1122 and the reference beam reflected from a reference mirror mounted on the lens housing 1106 indicates changes in the position of the stage relative to the radiation beam. Furthermore, in other embodiments the interferometry system 1126 can be positioned to measure changes in the position of reticle (or mask) stage 1116 or other movable components of the scanner system. Finally, the interferometry systems can be used in a similar fashion with lithography systems involving steppers, in addition to, or rather than, scanners.

As is well known in the art, lithography is a critical part of manufacturing methods for making semiconducting devices. For example, U.S. Pat. No. 5,483,343 outlines steps for such manufacturing methods. These steps are described below with reference to FIGS. 8 a and 8 b. FIG. 8 a is a flow chart of the sequence of manufacturing a semiconductor device such as a semiconductor chip (e.g., IC or LSI), a liquid crystal panel or a CCD. Step 1151 is a design process for designing the circuit of a semiconductor device. Step 1152 is a process for manufacturing a mask on the basis of the circuit pattern design. Step 1153 is a process for manufacturing a wafer by using a material such as silicon.

Step 1154 is a wafer process which is called a pre-process wherein, by using the so prepared mask and wafer, circuits are formed on the wafer through lithography. To form circuits on the wafer that correspond with sufficient spatial resolution those patterns on the mask, interferometric positioning of the lithography tool relative the wafer is necessary. The interferometry methods and systems described herein can be especially useful to improve the effectiveness of the lithography used in the wafer process.

Step 1155 is an assembling step, which is called a post-process wherein the wafer processed by step 1154 is formed into semiconductor chips. This step includes assembling (dicing and bonding) and packaging (chip sealing). Step 1156 is an inspection step wherein operability check, durability check and so on of the semiconductor devices produced by step 1155 are carried out. With these processes, semiconductor devices are finished and they are shipped (step 1157).

FIG. 8 b is a flow chart showing details of the wafer process. Step 1161 is an oxidation process for oxidizing the surface of a wafer. Step 1162 is a CVD process for forming an insulating film on the wafer surface. Step 1163 is an electrode forming process for forming electrodes on the wafer by vapor deposition. Step 1164 is an ion implanting process for implanting ions to the wafer. Step 1165 is a resist process for applying a resist (photosensitive material) to the wafer. Step 1166 is an exposure process for printing, by exposure (i.e., lithography), the circuit pattern of the mask on the wafer through the exposure apparatus described above. Once again, as described above, the use of the interferometry systems and methods described herein improve the accuracy and resolution of such lithography steps.

Step 1167 is a developing process for developing the exposed wafer. Step 1168 is an etching process for removing portions other than the developed resist image. Step 1169 is a resist separation process for separating the resist material remaining on the wafer after being subjected to the etching process. By repeating these processes, circuit patterns are formed and superimposed on the wafer.

The interferometry systems described above can also be used in other applications in which the relative position of an object needs to be measured precisely. For example, in applications in which a write beam such as a laser, x-ray, ion, or electron beam, marks a pattern onto a substrate as either the substrate or beam moves, the interferometry systems can be used to measure the relative movement between the substrate and write beam.

As an example, a schematic of a beam writing system 1200 is shown in FIG. 9. A source 1210 generates a write beam 1212, and a beam focusing assembly 1214 directs the radiation beam to a substrate 1216 supported by a movable stage 1218. To determine the relative position of the stage, an interferometry system 1220 directs a reference beam 1222 to a mirror 1224 mounted on beam focusing assembly 1214 and a measurement beam 1226 to a mirror 1228 mounted on stage 1218. Since the reference beam contacts a mirror mounted on the beam focusing assembly, the beam writing system is an example of a system that uses a column reference. Interferometry system 1220 can be any of the interferometry systems described previously. Changes in the position measured by the interferometry system correspond to changes in the relative position of write beam 1212 on substrate 1216. Interferometry system 1220 sends a measurement signal 1232 to controller 1230 that is indicative of the relative position of write beam 1212 on substrate 1216. Controller 1230 sends an output signal 1234 to a base 1236 that supports and positions stage 1218. In addition, controller 1230 sends a signal 1238 to source 1210 to vary the intensity of, or block, write beam 1212 so that the write beam contacts the substrate with an intensity sufficient to cause photophysical or photochemical change only at selected positions of the substrate.

Furthermore, in some embodiments, controller 1230 can cause beam focusing assembly 1214 to scan the write beam over a region of the substrate, e.g., using signal 1244. As a result, controller 1230 directs the other components of the system to pattern the substrate. The patterning is typically based on an electronic design pattern stored in the controller. In some applications the write beam patterns a resist coated on the substrate and in other applications the write beam directly patterns, e.g., etches, the substrate.

An important application of such a system is the fabrication of masks and reticles used in the lithography methods described previously. For example, to fabricate a lithography mask an electron beam can be used to pattern a chromium-coated glass substrate. In such cases where the write beam is an electron beam, the beam writing system encloses the electron beam path in a vacuum. Also, in cases where the write beam is, e.g., an electron or ion beam, the beam focusing assembly includes electric field generators such as quadrapole lenses for focusing and directing the charged particles onto the substrate under vacuum. In other cases where the write beam is a radiation beam, e.g., x-ray, UV, or visible radiation, the beam focusing assembly includes corresponding optics and for focusing and directing the radiation to the substrate.

A number of embodiments of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. Accordingly, other embodiments are within the scope of the following claims. 

1. A method comprising: directing a first portion of a beam including a first frequency component along a first path; frequency shifting a second portion of the beam to generate a shifted beam that includes a second frequency component different from the first frequency component and one or more spurious frequency components different from the first frequency component, wherein frequency shifting the second beam portion comprises modulating the phase of the second beam portion to generate the shifted beam, and the phase modulation comprises Serrodyne modulation; directing at least a portion of the shifted beam along a second path different from the first path; measuring an interference signal S(t) from interference between the beam portions directed along the different paths, wherein the signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the paths, where n is an average refractive index along the paths, {tilde over (L)}(t) is a total physical path difference between the paths, and t is time; and providing an error signal to reduce errors in an estimate of {tilde over (L)}(t) that are caused by at least one of the spurious frequency components of the shifted beam, the error signal being derived at least in part based on the signal S(t).
 2. The method of claim 1, further comprising providing the estimate of {tilde over (L)}(t) based at least in part on subtracting the error signal from the signal S(t).
 3. The method of claim 1, wherein the phase modulation comprises electro-optic phase modulation.
 4. The method of claim 1, wherein the errors are dependent on a non-zero fall time of a ramp associated with the Serrodyne modulation.
 5. The method of claim 4, wherein the errors are further dependent on an amplitude of the ramp.
 6. The method of claim 1, further comprising providing the estimate of {tilde over (L)}(t) based at least in part on adjusting the phase modulation of the second beam portion using the error signal.
 7. The method of claim 6, wherein the phase modulation is adjusted using the phase of the error signal.
 8. The method of claim 7, wherein an amplitude of the phase modulation is adjusted using the phase of the error signal.
 9. The method of claim 8, wherein adjusting the amplitude of the phase modulation using the phase of the error signal comprises adjusting the amplitude of the phase modulation away from an exact 2π phase shift of the second beam portion.
 10. The method of claim 9, wherein adjusting the amplitude of the phase modulation using the phase of the error signal comprises adjusting the amplitude of the phase modulation to be larger than an exact 2π phase shift of the second beam portion.
 11. The method of claim 1, wherein at least one of the beam portions is directed to reflect from a movable measurement object before producing the interference signal S(t) to generate a Doppler shift indicative of movement of the measurement object.
 12. The method of claim 11, further comprising providing the estimate of {tilde over (L)}(t) based at least in part on a sinusoidal component of the signal S(t) that has a time-varying argument corresponding to a sum of a term proportional to a difference between peak frequencies of the first frequency component and the second frequency component and a term proportional to a Doppler shift frequency.
 13. The method of claim 12, wherein the error signal includes a sinusoidal component that has a time-varying argument corresponding to a difference between the term proportional to the difference between the peak frequencies of the first frequency component and the second frequency component and the term proportional to the Doppler shift frequency.
 14. The method of claim 11, wherein the difference between the peak frequencies of the first frequency component and the second frequency component is less than about 500 kHz.
 15. The method of claim 14, wherein the Doppler shift frequency is less than about 50 kHz.
 16. The method of claim 1, wherein one of the paths is associated with a position of a reference object and the other path is associated with a position of a moveable measurement object.
 17. The method of claim 16, wherein the position of the moveable object is controlled by a servo system.
 18. The method of claim 17, wherein the servo system controls the position of the measurement object based on the signal S(t) and the error signal.
 19. The method of claim 1, wherein providing the error signal to reduce the errors caused by at least one spurious frequency component in the shifted beam comprises providing one or more coefficients representative of one or more errors that cause the signal S(t) to deviate from an ideal expression of the form A₁ cos(ω_(R)t+φ(t)+ζ₁), where A₁ and ζ₁ are constants, ω_(R) is an angular frequency difference between the first frequency component and the second frequency component, and φ(t)=nk{tilde over (L)}(t), with k=2π/λ and λ equal to a wavelength for the beam.
 20. The method of claim 19, wherein the deviation can be expressed as ${\sum\limits_{m,p}{A_{m,p}{\cos\left( {{\omega_{R}t} + {\frac{m}{p}{\varphi(t)}} + \zeta_{m,p}} \right)}}},$ where p=1, 2, 3 . . . , and m is any integer not equal to p, and where the provided coefficients comprise information corresponding to at least some of A_(m,p) and ζ_(m,p).
 21. The method of claim 19, wherein the coefficients are derived at least in part based on a plurality of samples of the signal S(t).
 22. The method of claim 21, wherein the error signal is generated from the coefficients and one or more error basis functions derived at least in part from a plurality of samples of the signal S(t).
 23. The method of claim 22, wherein each of the error basis functions is derived at least in part from a linear combination of samples of the signal S(t).
 24. The method of claim 22, wherein each of the error basis functions corresponds to a function that includes one or more leading sinusoidal terms having a time-varying argument that corresponds to a time-varying argument of an error term that represents a portion of the deviation of S(t) from the ideal expression.
 25. The method of claim 1, wherein reducing errors in the estimate of {tilde over (L)}(t) comprises deriving the estimate of {tilde over (L)}(t) from a difference between the error signal and a discrete Fourier transform of samples of S(t).
 26. A lithography method for use in fabricating integrated circuits on a wafer, the method comprising: supporting the wafer on a moveable stage; imaging spatially patterned radiation onto the wafer; adjusting the position of the stage; and monitoring the position of the stage using an interferometry system, wherein monitoring the position of the stage comprises reducing errors in an estimate of a physical path difference associated with a position of a measurement object associated with the stage using the method of claim
 1. 27. A method for fabricating integrated circuits, the method comprising: applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography method of claim 26; and producing an integrated circuit from the wafer.
 28. A lithography method for use in the fabrication of integrated circuits, the method comprising: directing input radiation through a mask to produce spatially patterned radiation; positioning the mask relative to the input radiation; monitoring the position of the mask relative to the input radiation using an interferometry system, wherein monitoring the position of the mask comprises reducing errors in an estimate of a physical path difference associated with the position of the mask using the method of claim 1; and imaging the spatially patterned radiation onto a wafer.
 29. A method for fabricating integrated circuits, the method comprising: applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography method of claim 28; and producing an integrated circuit from the wafer.
 30. A lithography method for fabricating integrated circuits on a wafer, the method comprising: positioning a first component of a lithography system relative to a second component of a lithography system to expose the wafer to spatially patterned radiation; and monitoring the position of the first component relative to the second component using an interferometry system, wherein monitoring the position of the first component comprises reducing errors in an estimate of a physical path difference associated with a position of a measurement object associated with the first component using the method of claim
 1. 31. The method of claim 30, wherein the first component comprises projection optics, and the second component comprises a stage for supporting the wafer.
 32. A method for fabricating integrated circuits, the method comprising: applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography method of claim 30; and producing an integrated circuit from the wafer.
 33. A method for fabricating a lithography mask, the method comprising: directing a write beam to a substrate to pattern the substrate; positioning the substrate relative to the write beam; and monitoring the position of the substrate relative to the write beam using an interferometry system, wherein monitoring the position of the substrate comprises reducing errors in an estimate of a physical path difference associated with a position of a measurement object associated with the substrate using the method of claim
 1. 34. A computer readable medium storing instructions that cause a processor to perform the method of claim
 1. 35. An apparatus comprising: an interferometry system, which during operation directs a first portion of a beam including a first frequency component along a first path; frequency shifts a second portion of the beam to generate a shifted beam that includes a second frequency component different from the first frequency component and one or more spurious frequency components different from the first frequency component, wherein the second beam portion is frequency shifted in a phase modulator that modulates the phase of the second beam portion to generate the shifted beam, and the phase modulation comprises Serrodyne modulation; and directs at least a portion of the shifted beam along a second path different from the first path; a detector that measures an interference signal S(t) from interference between the beam portions directed along the different paths, wherein the signal S(t) is indicative of changes in an optical path difference n{tilde over (L)}(t) between the paths, where n is an average refractive index along the paths, {tilde over (L)}(t) is a total physical path difference between the paths, and t is time; and an electronic processor, which during operation receives the interference signal S(t) from the detector and provides an error signal to reduce errors in an estimate of {tilde over (L)}(t) that are caused by at least one of the spurious frequency components of the shifted beam, the error signal being derived at least in part based on the signal S(t).
 36. The apparatus of claim 35, wherein the phase modulator comprises an electro-optic phase modulator.
 37. A lithography system for use in fabricating integrated circuits on a wafer, the system comprising: a stage for supporting the wafer; an illumination system for imaging spatially patterned radiation onto the wafer; a positioning system for adjusting the position of the stage relative to the imaged radiation; and the apparatus of claim 35 for monitoring the position of the wafer relative to the imaged radiation.
 38. A method for fabricating integrated circuits, the method comprising: applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography system of claim 37; and producing an integrated circuit from the wafer.
 39. A lithography system for use in fabricating integrated circuits on a wafer, the system comprising: a stage for supporting the wafer; and an illumination system including a radiation source, a mask, a positioning system, a lens assembly, and the apparatus of claim 35, wherein during operation the source directs radiation through the mask to produce spatially patterned radiation, the positioning system adjusts the position of the mask relative to the radiation from the source, the lens assembly images the spatially patterned radiation onto the wafer, and the apparatus monitors the position of the mask relative to the radiation from the source.
 40. A method for fabricating integrated circuits, the method comprising: applying a resist to a wafer; forming a pattern of a mask in the resist by exposing the wafer to radiation using the lithography system of claim 39; and producing an integrated circuit from the wafer.
 41. A beam writing system for use in fabricating a lithography mask, the system comprising: a source providing a write beam to pattern a substrate; a stage supporting the substrate; a beam directing assembly for delivering the write beam to the substrate; a positioning system for positioning the stage and beam directing assembly relative one another; and the apparatus of claim 35 for monitoring the position of the stage relative to the beam directing assembly.
 42. A method for fabricating a lithography mask comprising: directing a beam to a substrate using the beam writing system of claim 41; varying the intensity or the position of the beam at the substrate to form a pattern in the substrate; and forming the lithography mask from the patterned substrate. 